Daily Papers

We put forward the dynamical study of a novel higher-order small network of Chialvo neurons arranged in a ring-star topology, with the neurons interacting via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through higher-order multi-body interactions. We first analyze our model using the tools from nonlinear dynamics literature: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of chaotic attractors, and also an intriguing route to chaos starting from a fixed point, to period-doubling, to cyclic quasiperiodic closed invariant curves, to ultimately chaos. We numerically observe the existence of codimension-1 bifurcation patterns: saddle-node, period-doubling, and Neimark Sacker. We also qualitatively study the typical phase portraits of the system and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure respectively. Finally, we study the collective behavior of the neurons in terms of two synchronization measures: the cross-correlation coefficient, and the Kuramoto order parameter.

We present a numerical framework to treat infinite time horizon spatially distributed optimal control problems via the associated canonical system derived by Pontryagin's Maximum Principle. The basic idea is to consider the canonical system in two steps. First we perform a bifurcation analysis of canonical steady states using the continuation and bifurcation package pde2path, yielding a number of so called flat and patterned canonical steady states. In a second step we link pde2path to the two point boundary value problem solver TOM to study time dependent canonical paths to steady states having the so called saddle point property. As an example we consider a shallow lake model with diffusion.

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.

Traveling wavetrains in generalized two-species predator-prey models and two-component reaction-diffusion equations are considered. The stability of the fixed points of the traveling wave ODEs (in the usual "spatial" variable) is considered. For general functional forms of the nonlinear prey birthrate/prey deathrate or reaction terms, a Hopf bifurcation is shown to occur at two different critical values of the traveling wave speed. The post-bifurcation dynamics is investigated for five different functional forms of the nonlinearities. In cases where the bifurcation is supercritical, the post-bifurcation behaviour yields stable periodic orbits of the traveling-wave ODEs in the spatial variable. These correspond to stable periodic wavetrains of the full PDEs. Subcritical Hopf bifurcations yield more complex post-bifurcation dynamics in the PDE wavetrains. In special cases where the subcritical bifurcation marks the end of the regime of stability, the post-bifurcation behavior in the spatial ODEs is chaotic, corresponding to wavetrains of the original PDEs which are spatially coherent, but have chaotic temporal dynamics. All the models are integrated numerically to investigate the post-bifurcation dynamics and chaotic regimes are characterized by computing power spectra, autocorrelation functions, and fractal dimensions.

Singularity Theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg-Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcations curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.

We study the one-dimensional nonlocal Kirchhoff type bifurcation problem related to logistic equation of population dynamics. We establish the precise asymptotic formulas for bifurcation curve lambda = lambda(alpha) as alpha to infty in L^2-framework, where alpha:= Vert u_lambda Vert_2.

This work investigates the number of limit cycles bifurcating from the periodic orbits of a linear center when it is perturbed by piecewise holomorphic systems with three zones separated by two external circles. We analyze how different geometric configurations of these zones influence the number of limit cycles. Moreover, we show the existence of at least sixteen limit cycles in fourth-degree piecewise polynomial holomorphic systems under certain configurations.

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the rightmost, stability-determining roots of the characteristic equation which can subsequently be used as starting values in a Newton procedure. For periodic solutions, approximations to the Floquet multipliers are computed. The manual describes the structure of the package, its routines, and its data and method parameter structures.

In this paper we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate the periodic solutions of the complex cubic-quintic Ginzburg-Landau equation. The primary tools used here are Hopf bifurcation theory and perturbation theory. Explicit results are obtained for the post-bifurcation periodic orbits and their stability. Generalized and degenerate Hopf bifurcations are also briefly considered to track the emergence of global structure such as homoclinic orbits.

All known up to now models of chemical oscillations are based exclusively on kinetic considerations. The chemical gross-process equation is split usually by elementary steps, each step is supplied by an arrow and a differential equation, joint solution to such a construction under certain, often ad hoc chosen conditions and with ad hoc numerical coefficients leads to chemical oscillations. Kinetic perception of chemical oscillations reigns without exclusions. However, as it was recently shown by the author for the laser and for the electrochemical systems, chemical oscillations follow also from solutions to the basic expressions of discrete thermodynamics of chemical equilibria. Graphically those solutions are various fork bifurcation diagrams, and, in certain types of chemical systems, oscillations are well pronounced in the bistable bifurcation areas. In this work we describe a general thermodynamic approach to chemical oscillations as opposite to kinetic models, and depict some of their new features like spontaneity and fractality. The paper doubts exclusivity of the kinetic approach to chemical oscillations, and its aim is to discuss and exemplify thermodynamically predicted chemical oscillations in closed chemical systems.

The realization that complex systems such as ecological communities can collapse or shift regimes suddenly and without rapid external forcing poses a serious challenge to our understanding and management of the natural world. The potential to identify early warning signals that would allow researchers and managers to predict such events before they happen has therefore been an invaluable discovery that offers a way forward in spite of such seemingly unpredictable behavior. Research into early warning signals has demonstrated that it is possible to define and detect such early warning signals in advance of a transition in certain contexts. Here we describe the pattern emerging as research continues to explore just how far we can generalize these results. A core of examples emerges that shares three properties: the phenomenon of rapid regime shifts, a pattern of 'critical slowing down' that can be used to detect the approaching shift, and a mechanism of bifurcation driving the sudden change. As research has expanded beyond these core examples, it is becoming clear that not all systems that show regime shifts exhibit critical slowing down, or vice versa. Even when systems exhibit critical slowing down, statistical detection is a challenge. We review the literature that explores these edge cases and highlight the need for (a) new early warning behaviors that can be used in cases where rapid shifts do not exhibit critical slowing down, (b) the development of methods to identify which behavior might be an appropriate signal when encountering a novel system; bearing in mind that a positive indication for some systems is a negative indication in others, and (c) statistical methods that can distinguish between signatures of early warning behaviors and noise.

We propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.

Neural networks acquire structured representations at specific moments during training, yet identifying these transitions typically relies on retrospective, label-dependent metrics. We introduce a bifurcation theory of representation dynamics to detect these moments in real time. Analyzing a passive GMM probe attached to the evolving encoder, we show the onset of structure corresponds to a supercritical pitchfork bifurcation driven by the loss Hessian. The system exhibits a theoretically predictable zero-crossing (β_c) that, compared to the network's current state (β), yields a dynamic ratio β(t)/β_c(t): a universal, label-free phase coordinate for representation dynamics, computable entirely from hidden states. We empirically validate four distinct transition regimes predicted by this coordinate across diverse settings: SAEs on language models (Pythia), SSL (CIFAR), and grokking (modular arithmetic). Crucially, under finite dissipation, macroscopic symmetry-breaking can lag the initial zero-crossing by orders of magnitude, which providing a rigorous dynamical account of the delayed escape observed in grokking. Microscopically, the bifurcation creates a shared unstable subspace, forcing collective symmetry breaking. We term this the "feature lottery" in SAE training: a feature's terminal interpretability becomes predictable remarkably early. By only 5% of training, early atom purity robustly predicts final convergence purity, with top-decile early atoms achieving over 12x the baseline purity at convergence. Beyond explaining concept emergence, β/β_c provides a practical early-warning indicator for training health, detecting the onset of usable structure, the crystallization of feature identity, and representational collapse epochs before downstream metrics react.

We study a three-species competitive Lotka--Volterra system with cyclic dominance, motivated by rock--paper--scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order 1/ε, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime 0<εll 1. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed of three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weierstrass wp functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of wp functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves.

Rate-induced tipping ("R-tipping") in ecological modelling is characterised by too-rapid change of an environmental parameter causing collapse of a population without any bifurcation being crossed ("B-tipping"). A well-known example of a slow-fast predator-prey model in which R-tipping is observed [Vanselow, Wieczorek, Feudel, Journal of Theoretical Biology, 479, 64-72 (2019)] suffers from unecological "resurgence", whereby populations driven to functional extinction recover towards stable coexistence. We propose an analytically tractable model, incorporating a strong Allee effect into the prey dynamics in a slow-fast Lotka-Volterra-type predator-prey system, which induces bistability and thus renders the extinction state a genuine attractor. Applying geometric singular perturbation theory (GSPT), we describe the dynamics of the extended model that is obtained by "ramping" of the inverse prey carrying capacity. We show the presence of a parabolic-shaped, folded critical manifold which admits a folded saddle singularity, the strong canard of which separates solutions that track the moving coexistence state from those that tip to extinction. The analytical simplicity of our model allows us to derive explicit expressions for the critical rate that separates "tracking" from "tipping" dynamics. Finally, we argue that our system represents a minimal model for rate-induced tipping to extinction, in that it incorporates three essential ingredients: bistability, a folded critical manifold, and a folded-saddle-type canard.

We introduce rd-spiral, an open-source Python library for simulating 2D reaction-diffusion systems using pseudo-spectral methods. The framework combines FFT-based spatial discretization with adaptive Dormand-Prince time integration, achieving exponential convergence while maintaining pedagogical clarity. We analyze three dynamical regimes: stable spirals, spatiotemporal chaos, and pattern decay, revealing extreme non-Gaussian statistics (kurtosis >96) in stable states. Information-theoretic metrics show 10.7% reduction in activator-inhibitor coupling during turbulence versus 6.5% in stable regimes. The solver handles stiffness ratios >6:1 with features including automated equilibrium classification and checkpointing. Effect sizes (delta=0.37--0.78) distinguish regimes, with asymmetric field sensitivities to perturbations. By balancing computational rigor with educational transparency, rd-spiral bridges theoretical and practical nonlinear dynamics.

Using a variational formulation for partial differential equations (PDEs) combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE are varied. Our predictions on the variation of the soliton amplitude, width, position, speed and phase of the solutions using the variational formulation agree with simulation results.

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of BBMv experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.

Regulatory institutions (from content moderation platforms to financial supervisors) observe, deliberate, and intervene only after a characteristic delay. We ask whether this processing lag alone can destabilize a multi-agent system that would otherwise remain stable, without exogenous shocks, coordination among agents, or malicious actors. We study this question in two stages. First, we analyze a delayed replicator equation in which autonomous agents receive a benefit from radical behavior but face punishment based on a lagged institutional alarm signal. We derive a closed-form critical delay threshold beyond which the unique interior equilibrium loses stability through a Hopf bifurcation, and prove via center manifold reduction that the bifurcation is supercritical (producing bounded oscillations, not explosive growth) for the entire sigmoid response-function family. Second, we embed N=240 agents on a network and equip them with reinforcement learning (tabular Q-learning), comparing three decision architectures in a factorial design: non-reactive agents (fixed policy), reactive agents (threshold heuristic without memory), and Q-learning agents (adaptive with cumulative value estimates). The results reveal a hierarchy opposite to the naive expectation that learning amplifies instability: non-reactive agents are immune to delay (0% runaway across all tested values), reactive agents collapse catastrophically (96% runaway by delay geq 8 steps), and Q-learning agents achieve partial resilience (66% runaway at delay = 20). The destabilizing ingredient is reactivity to delayed signals: agents that immediately exploit low-alarm windows trigger oscillatory feedback loops. Learning buffers this through implicit punishment memory encoded in Q-values

Critical transitions - abrupt, often irreversible changes in system dynamics - arise across human and natural systems, often with catastrophic consequences. Real-world observations of such shifts remain scarce, preventing the development of reliable early warning systems. Conventional statistical and spectral indicators, such as increasing variance, tend to fail under realistic conditions of limited data and correlated noise, whereas existing deep learning classifiers do not extrapolate beyond their training data distribution. In this work, we introduce TipPFN, an in-context learning (ICL) framework that uses a prior-data fitted network to infer a system's proximity to a critical transition. Trained on our novel synthetic data generator, which is based on canonical bifurcation scenarios coupled to diverse, randomized stochastic dynamics, TipPFN flexibly capitalizes on contexts of various sizes, complexity and dimensionalities. We demonstrate robust, state-of-the-art early detection of critical transitions in previously unseen tipping regimes, sim-to-real examples, and real-world observations in both ICL and zero-shot settings.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a continuous piecewise linear function with exactly \(n\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(n\) when \(F\) has exactly \(n\) fold points. The conjecture was confirmed for \(n=1\) and \(n=2\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \(n\ge3\) remained open. In this paper, we show that the maximal number of limit cycles is at least \(|3n-4|\) for \(n\in\mathbb N^+\), thereby disproving Tonnelier's conjecture for \(n\ge3\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \(2^{28(n+1)^2}\).

This chapter provides a pedagogical introduction and overview of spatial and temporal correlation and fluctuation effects resulting from the fundamentally stochastic kinetics underlying chemical reactions and the dynamics of populations or epidemics. After reviewing the assumptions and mean-field type approximations involved in the construction of chemical rate equations for uniform reactant densities, we first discuss spatial clustering in birth-death systems, where non-linearities are introduced through either density-limiting pair reactions, or equivalently via local imposition of finite carrying capacities. The competition of offspring production, death, and non-linear inhibition induces a population extinction threshold, which represents a non-equilibrium phase transition that separates active from absorbing states. This continuous transition is characterized by the universal scaling exponents of critical directed percolation clusters. Next we focus on the emergence of depletion zones in single-species annihilation processes and spatial population segregation with the associated reaction fronts in two-species pair annihilation. These strong (anti-)correlation effects are dynamically generated by the underlying stochastic kinetics. Finally, we address noise-induced and fluctuation-stabilized spatio-temporal patterns in basic predator-prey systems, exemplified by spreading activity fronts in the two-species Lotka-Volterra model as well as spiral structures in the May-Leonard variant of cyclically competing three-species systems akin to rock-paper-scissors games.

We develop a dynamical mean-field theory for random recurrent networks with low-rank structure and firing-rate-driven adaptation. When the random connectivity is strong enough to generate chaos, increasing adaptation strength drives the network through four regimes: a static coherent state, noise-sustained oscillations that progress from regular to irregular, stochastic switching between symmetric wells, and a global limit cycle. The theory identifies two instability mechanisms, chaos onset from the random connectivity and a Hopf bifurcation of the coherent mode, and shows how adaptation shapes both through the frequency-dependent single-neuron transfer function. A reduced three-dimensional model captures the bifurcation structure of the full network. Above the chaos threshold, coherent population-level oscillations coexist with heterogeneous firing rates and network-generated stochasticity at the single-neuron level. The interaction of adaptation with random and low-rank connectivity produces a rich oscillatory repertoire, including waxing-and-waning rhythmic episodes, persistent state switching, and slow Up-Down alternations, dynamics that have been observed during wakefulness, sleep, and anesthesia.

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in R^3 whose positive phases are periodic perturbations of the product of the Hauswirth--Hélein--Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in R^{n+1}, n geq 3, bifurcating from product solutions U(x,t)=u(x), when the base u:R^nto [0,infty) has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions nin{3,4}, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in R^{4} and R^{5} bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

Ecological interactions are often mediated by feedbacks between organisms and their resource environments. Yet, how resource supply dynamics dictate collective dynamical phases of an ecosystem remains unclear. Here, we analyse a generalised consumer--resource model with non-reciprocal interactions to demonstrate that self-renewing versus externally-supplied resources yield fundamentally different dynamical phase diagrams. As interactions become increasingly non-reciprocal, ecosystems relying on self-renewing resources transition from stable dynamics to chaos and ultimately to infeasibility. By contrast, ecosystems with externally-supplied resources remain stable over a broader parameter range and transition to infeasibility without experiencing an intervening chaotic phase. Using the cavity method, we derive a unified stability condition applicable to a broad class of resource supply functions, explaining why externally-supplied resources can expand the stable region. We show that stability hinges crucially on the susceptibility of resources to perturbations, which depends strongly on their supply. Further, we show that external resource supply suppresses chaos in the unstable region by drastically reducing the susceptibility of resources closest to extinction. Our findings demonstrate that resource dynamics fundamentally reshape the accessible dynamical behaviours of an ecosystem, with implications for interpreting microbial community experiments.

The transition of an impulsively excited kink oscillation of a solar coronal loop to an oscillation with a stationary amplitude, i.e., the damping pattern, is determined using the low-dimensional self-oscillation model. In the model, the decayless kink oscillations are sustained by the interaction of the oscillating loop with an external quasi-steady flow. The analytical solution is based on the assumption that the combined effect of the effective dissipation, for example, by resonant absorption, and interaction with an external flow, is weak. The effect is characterised by a dimensionless coupling parameter. The damping pattern is found to depend upon the initial amplitude and the coupling parameter. The approximate expression shows a good agreement with a numerical solution of the self-oscillation equation. The plausibility of the established damping pattern is demonstrated by an observational example. Notably, the damping pattern is not exponential, and the characteristic decay time is different from the time determined by the traditionally used exponential damping fit. Implications of this finding for seismology of the solar coronal plasmas are discussed. In particular, it is suggested that a very rapid, in less than the oscillation period, decay of the oscillation to the stationary level, achieved for larger values of the coupling parameter, can explain the relative rareness of the kink oscillation events.

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $ U_x = P, qquad P_x = - f(U) - βP + tilde α(x)U, arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the L^1-norm of the control \tilde α are determined by the unstable and stable manifolds Γ_u, Γ_s, the critical curve P^*(U) = Uf(U), and the associated function F(U) = dP^*{dU} + f(U){P^*(U)}. We establish generic transversality properties for these objects with respect to perturbations of the speed β and the nonlinear source f, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed β^{**}, where the unstable manifold approaches the strongly stable manifold of the equilibrium (U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

We analyze a class of cell-bulk coupled PDE-ODE models, motivated by quorum and diffusion sensing phenomena in microbial systems, that characterize communication between localized spatially segregated dynamically active signaling compartments that have a permeable boundary. Each cell secretes a signaling chemical into the bulk region at a constant rate and receives a feedback of the bulk chemical from the entire collection of cells. This global feedback, which activates signaling pathways within the cells, modifies the intracellular dynamics according to the external environment. The cell secretion and global feedback are regulated by permeability parameters across the cell membrane. For arbitrary reaction-kinetics within each cell, the method of matched asymptotic expansions is used in the limit of small cell radius to construct steady-state solutions of the PDE-ODE model, and to derive a globally coupled nonlinear matrix eigenvalue problem (GCEP) that characterizes the linear stability properties of the steady-states. In the limit of large bulk diffusivity an asymptotic analysis of the PDE-ODE model leads to a limiting ODE system for the spatial average of the concentration in the bulk region that is coupled to the intracellular dynamics within the cells. Results from the linear stability theory and ODE dynamics are illustrated for Sel'kov reaction-kinetics, where the kinetic parameters are chosen so that each cell is quiescent when uncoupled from the bulk medium. For various specific spatial configurations of cells, the linear stability theory is used to construct phase diagrams in parameter space characterizing where a switch-like emergence of intracellular oscillations can occur through a Hopf bifurcation.

Economic activities favor mutual geographical proximity and concentrate spatially to form cities. In a world of diminishing transport costs, however, the advantage of physical proximity is fading, and the role of cities in the economy may be declining. To provide insights into the long-run evolution of cities, we analyzed Japan's census data over the 1970--2015 period. We found that fewer and larger cities thrived at the national scale, suggesting an eventual mono-centric economy with a single megacity; simultaneously, each larger city flattened out at the local scale, suggesting an eventual extinction of cities. We interpret this multi-scale phenomenon as an instance of pattern formation by self-organization, which is widely studied in mathematics and biology. However, cities' dynamics are distinct from mathematical or biological mechanisms because they are governed by economic interactions mediated by transport costs between locations. Our results call for the synthesis of knowledge in mathematics, biology, and economics to open the door for a general pattern formation theory that is applicable to socioeconomic phenomena.

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. Here, we extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, we derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. We show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics.

Chemical reaction networks can undergo nonequilibrium phase transitions upon variation of external control parameters like the chemical potential of a species. We investigate the flux in the associated chemostats that is proportional to the entropy production and its critical fluctuations within the Schlögl model. Numerical simulations show that the corresponding diffusion coefficient diverges at the critical point as a function of system size. In the vicinity of the critical point, the diffusion coefficient follows a scaling form. We develop an analytical approach based on the chemical Langevin equation and van Kampen's system size expansion that yields the corresponding exponents in the monostable regime. In the bistable regime, we rely on a two-state approximation in order to analytically describe the critical behavior.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

In this study, we propose a multi branched network approach to predict the dynamics of a physics attractor characterized by intricate and chaotic behavior. We introduce a unique neural network architecture comprised of Radial Basis Function (RBF) layers combined with an attention mechanism designed to effectively capture nonlinear inter-dependencies inherent in the attractor's temporal evolution. Our results demonstrate successful prediction of the attractor's trajectory across 100 predictions made using a real-world dataset of 36,700 time-series observations encompassing approximately 28 minutes of activity. To further illustrate the performance of our proposed technique, we provide comprehensive visualizations depicting the attractor's original and predicted behaviors alongside quantitative measures comparing observed versus estimated outcomes. Overall, this work showcases the potential of advanced machine learning algorithms in elucidating hidden structures in complex physical systems while offering practical applications in various domains requiring accurate short-term forecasting capabilities.

The paper sets forth comprehensive basics of Discrete Thermodynamics of Chemical Equilibria (DTD), developed by the author during the last decade and spread over series of publications. Based on the linear equations of irreversible thermodynamics, De Donder's definition of the thermodynamic force, and the Le Chatelier principle, DTD brings forward a notion of chemical equilibrium as a balance of internal and external thermodynamic forces, acting against a chemical system. The basic expression of DTD is a logistic map that ties together energetic characteristics of the chemical transformation in the system, its deviation from true thermodynamic equilibrium, and the sum of thermodynamic forces, causing that deviation. System deviation from thermodynamic equilibrium is the major variable of the theory. Solutions to the basic map define the chemical system domain of states comprising bifurcation diagrams with four areas, from true thermodynamic equilibrium to chaos, having specific distinctive meaning for chemical systems. The theory is derived from the currently recognized ideas of chemical thermodynamics and binds classical and contemporary thermodynamics of chemical equilibria into a unique concept. DTD opens new opportunities in understanding and analysis of equilibria in chemical systems. Some new results, included in the paper, have never been published before.

We present a concise dynamical picture of infant-driven household chaos. The framework has three postulations: recurrent daily chaos, overall entropy growth in household organization, and transient local ordering episodes with switching rules (a volatile Maxwell-demon effect). We illustrate entropy growth with a two-region toy model (organizer vs. play area), where entropy production is nonnegative and long-time behavior is typically play-area dominated. We also model toy diffusion and curiosity-driven behavior, where novelty matters more than punishment in the short term, while gradual learning still occurs.

Physiological time series signals reflect complex, multi-scale dynamical processes of the human body. Existing modeling studies focus on static tasks such as classification, event forecasting, or short-horizon next step prediction, while long-horizon signal-level forecasting and predictive nature of physiological signals remain underexplored. We introduce NormWear-2, a world model that encodes both multivariate physiological signals and clinical intervention variables into a shared latent space and models their joint temporal evolution as a dynamical system. Our approach combines inference from prior pre-trained knowledge (intuition) with instant non-parametric latent state transition adaptation (insight), enabling coherent forecasting across multiple temporal scales, conditioned on heterogeneous clinical interventions. During the pretraining phase, we find that chaos-theoretic balancing of dynamical regime diversity yields more robust representations, with a smaller balanced corpus outperforming one twice its size and capturing bifurcation regimes. We evaluate the world model performance across diverse real-world physiological datasets spanning heterogeneous temporal resolutions and intervention regimes, covering daily life, point-of-care, and clinical settings, including fitness planning, hemodialysis, diabetes management, and surgical monitoring. These evaluation datasets comprise records from 8,026 subjects, spanning study durations from 3.2 hours for high-resolution signal data to 2.3 years for longitudinal clinical biomarker tracking. NormWear-2 achieves the best overall forecasting performance across time, frequency, and latent representation domains, with significant improvements over state-of-the-art time series foundation models, while maintaining competitive downstream representation quality, providing a step toward general-purpose world models for physiological signals.

A cumulative variational framework for solid deformation is proposed. Its central organizing principle is the recursive combination of three structural components of one material body: a compatible Cosserat component, a distributed-incompatibility component, and a defect-source component. At any prescribed external condition these components constitute a combined material state. They are governed under common physical boundary and initial conditions and are coupled through one cumulative energy and the associated field equations. They are not separate boundary-value problems and need not appear in a prescribed temporal sequence. The cumulative character of the framework concerns the response of the combined state to variation of externally prescribed conditions. For the special case of an initially perfect body, loss of stability may produce successive branches with defect and then source participation. Such a sequence is a bifurcation property of the selected initial state and constitutive energy, rather than a defining hierarchy of the framework. A one-dimensional specialization illustrates structural bifurcation, residual deformation and residual stress produced by retained structure, dependence on the initial combined state, directional memory under reverse loading, parametric cumulative hysteresis, and changes of observable tangent stiffness associated with structural stabilization. A spatial extension introduces defect and source gradient terms and yields a size-dependent structural stability operator. For the stable quartic energy considered, the gradient terms regularize nonuniform perturbations but do not by themselves generate a finite-wavelength localization instability on an already stable three-component branch. The examples thereby distinguish consequences of the cumulative structure and energy from phenomena requiring additional constitutive evolution laws.

Numerical simulations of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal five entirely novel classes of pulse or solitary waves solutions, viz. pulsating, creeping, snaking, erupting, and chaotical solitons. Here, we develop a theoretical framework for analyzing the full spatio-temporal structure of one class of dissipative solution (snaking soliton) of the CCQGLE using the variational approximation technique and the dynamical systems theory. The qualitative behavior of the snaking soliton is investigated using the numerical simulations of (a) the full nonlinear complex partial differential equation and (b) a system of three ordinary differential equations resulting from the variational approximation.

Generative Adversarial Networks (GANs) are a popular formulation to train generative models for complex high dimensional data. The standard method for training GANs involves a gradient descent-ascent (GDA) procedure on a minimax optimization problem. This procedure is hard to analyze in general due to the nonlinear nature of the dynamics. We study the local dynamics of GDA for training a GAN with a kernel-based discriminator. This convergence analysis is based on a linearization of a non-linear dynamical system that describes the GDA iterations, under an isolated points model assumption from [Becker et al. 2022]. Our analysis brings out the effect of the learning rates, regularization, and the bandwidth of the kernel discriminator, on the local convergence rate of GDA. Importantly, we show phase transitions that indicate when the system converges, oscillates, or diverges. We also provide numerical simulations that verify our claims.

Dynamical systems (DS) theory is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS separated by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and R\"ossler systems, AL-RNNs discover, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

We derive an expression for the variation between parallel trajectories in phenotypic evolution, extending the well known result that predicts the mean evolutionary path in adaptive dynamics or quantitative genetics. We show how this expression gives rise to the notion of fluctuation domains - parts of the fitness landscape where the rate of evolution is very predictable (due to fluctuation dissipation) and parts where it is highly variable (due to fluctuation enhancement). These fluctuation domains are determined by the curvature of the fitness landscape. Regions of the fitness landscape with positive curvature, such as adaptive valleys or branching points, experience enhancement. Regions with negative curvature, such as adaptive peaks, experience dissipation. We explore these dynamics in the ecological scenarios of implicit and explicit competition for a limiting resource.

Ridge Rider (RR) is an algorithm for finding diverse solutions to optimization problems by following eigenvectors of the Hessian ("ridges"). RR is designed for conservative gradient systems (i.e., settings involving a single loss function), where it branches at saddles - easy-to-find bifurcation points. We generalize this idea to non-conservative, multi-agent gradient systems by proposing a method - denoted Generalized Ridge Rider (GRR) - for finding arbitrary bifurcation points. We give theoretical motivation for our method by leveraging machinery from the field of dynamical systems. We construct novel toy problems where we can visualize new phenomena while giving insight into high-dimensional problems of interest. Finally, we empirically evaluate our method by finding diverse solutions in the iterated prisoners' dilemma and relevant machine learning problems including generative adversarial networks.

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.

Predicting the intermediate trajectories between an initial and target distribution is a central problem in generative modeling. Existing approaches, such as flow matching and Schr\"odinger Bridge Matching, effectively learn mappings between two distributions by modeling a single stochastic path. However, these methods are inherently limited to unimodal transitions and cannot capture branched or divergent evolution from a common origin to multiple distinct outcomes. To address this, we introduce Branched Schr\"odinger Bridge Matching (BranchSBM), a novel framework that learns branched Schr\"odinger bridges. BranchSBM parameterizes multiple time-dependent velocity fields and growth processes, enabling the representation of population-level divergence into multiple terminal distributions. We show that BranchSBM is not only more expressive but also essential for tasks involving multi-path surface navigation, modeling cell fate bifurcations from homogeneous progenitor states, and simulating diverging cellular responses to perturbations.

The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying attractor. Chaotic systems thus pose a unique challenge to modern statistical learning techniques, while retaining quantifiable mathematical properties that make them controllable and interpretable as benchmarks. Here, we present a growing database currently comprising 131 known chaotic dynamical systems spanning fields such as astrophysics, climatology, and biochemistry. Each system is paired with precomputed multivariate and univariate time series. Our dataset has comparable scale to existing static time series databases; however, our systems can be re-integrated to produce additional datasets of arbitrary length and granularity. Our dataset is annotated with known mathematical properties of each system, and we perform feature analysis to broadly categorize the diverse dynamics present across the collection. Chaotic systems inherently challenge forecasting models, and across extensive benchmarks we correlate forecasting performance with the degree of chaos present. We also exploit the unique generative properties of our dataset in several proof-of-concept experiments: surrogate transfer learning to improve time series classification, importance sampling to accelerate model training, and benchmarking symbolic regression algorithms.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

Time series play a fundamental role in many domains, capturing a plethora of information about the underlying data-generating processes. When a process generates multiple synchronized signals we are faced with multidimensional time series. In this context a fundamental problem is that of motif mining, where we seek patterns repeating twice with minor variations, spanning some of the dimensions. State of the art exact solutions for this problem run in time quadratic in the length of the input time series. We provide a scalable method to find the top-k motifs in multidimensional time series with probabilistic guarantees on the quality of the results. Our algorithm runs in time subquadratic in the length of the input, and returns the exact solution with probability at least 1-δ, where δ is a user-defined parameter. The algorithm is designed to be adaptive to the input distribution, self-tuning its parameters while respecting user-defined limits on the memory to use. Our theoretical analysis is complemented by an extensive experimental evaluation, showing that our algorithm is orders of magnitude faster than the state of the art.

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.

This paper reports on patterns exhibiting self-replication with spontaneous, inheritable mutations and exponential genetic drift in Neural Cellular Automata. Despite the models not being explicitly trained for mutation or inheritability, the descendant patterns exponentially drift away from ancestral patterns, even when the automaton is deterministic. While this is far from being the first instance of evolutionary dynamics in a cellular automaton, it is the first to do so by exploiting the power and convenience of Neural Cellular Automata, arguably increasing the space of variations and the opportunity for Open Ended Evolution.

Dynamical systems with complex behaviours, e.g. immune system cells interacting with a pathogen, are commonly modelled by splitting the behaviour into different regimes, or modes, each with simpler dynamics, and then learning the switching behaviour from one mode to another. Switching Dynamical Systems (SDS) are a powerful tool that automatically discovers these modes and mode-switching behaviour from time series data. While effective, these methods focus on independent objects, where the modes of one object are independent of the modes of the other objects. In this paper, we focus on the more general interacting object setting for switching dynamical systems, where the per-object dynamics also depends on an unknown and dynamically changing subset of other objects and their modes. To this end, we propose a novel graph-based approach for switching dynamical systems, GRAph Switching dynamical Systems (GRASS), in which we use a dynamic graph to characterize interactions between objects and learn both intra-object and inter-object mode-switching behaviour. We introduce two new datasets for this setting, a synthesized ODE-driven particles dataset and a real-world Salsa Couple Dancing dataset. Experiments show that GRASS can consistently outperforms previous state-of-the-art methods.

Motivated by the peculiar features of the trans-Neptunian object (TNO) Máni and the discovery of ring systems around Centaurs and trans-Neptunian bodies, we investigate the stability of particles around a small irregular body hosting a deep equatorial crater. This study is the third contribution in a sequence devoted to the dynamics around non-spherically symmetric bodies. Using Máni as a reference model, whose crater depth exceeds 10% of its radius, we map the system stability through complementary methods: Poincaré surfaces of section (PSS), survival maps, and the finite-time Lyapunov exponent (FTLE). In the nominal case, the 1!:!1, 2!:!1, 3!:!1, and 4!:!1 spin-orbit resonances (SORs) are consistently identified by all techniques. The prominent 3!:!1 SOR exhibits a bifurcated structure at high eccentricities, matching structural transitions observed at lower Jacobi constant values in the PSSs. Increasing the rotation rate (λ) and crater mass ratio (μ) enlarges resonance widths and promotes overlap, making the 2!:!1 and 4!:!1 SORs progressively dominant. Lower rotation rates allow stable trajectories to persist at higher eccentricities, whereas the 1!:!1 SOR is highly sensitive to stronger perturbations and disappears in the most extreme cases. In contrast to mass-anomaly models, which can clear the region interior to the 2!:!1 SOR, the crater configuration considered here preserves stable regions in this vicinity, suggesting alternative scenarios for the formation and maintenance of debris structures around irregular minor bodies.

Differentiable programming techniques are widely used in the community and are responsible for the machine learning renaissance of the past several decades. While these methods are powerful, they have limits. In this short report, we discuss a common chaos based failure mode which appears in a variety of differentiable circumstances, ranging from recurrent neural networks and numerical physics simulation to training learned optimizers. We trace this failure to the spectrum of the Jacobian of the system under study, and provide criteria for when a practitioner might expect this failure to spoil their differentiation based optimization algorithms.

Many events occur in the world. Some event types are stochastically excited or inhibited---in the sense of having their probabilities elevated or decreased---by patterns in the sequence of previous events. Discovering such patterns can help us predict which type of event will happen next and when. We model streams of discrete events in continuous time, by constructing a neurally self-modulating multivariate point process in which the intensities of multiple event types evolve according to a novel continuous-time LSTM. This generative model allows past events to influence the future in complex and realistic ways, by conditioning future event intensities on the hidden state of a recurrent neural network that has consumed the stream of past events. Our model has desirable qualitative properties. It achieves competitive likelihood and predictive accuracy on real and synthetic datasets, including under missing-data conditions.

By using ZEUS cluster at Embry-Riddle Aeronautical University we perform extensive numerical simulations based on a two-dimensional Fourier spectral method Fourier spatial discretization and an explicit scheme for time differencing) to find the range of existence of the spatiotemporal solitons of the two-dimensional complex Ginzburg-Landau equation with cubic and quintic nonlinearities. We start from the parameters used by Akhmediev {\it et. al.} and slowly vary them one by one to determine the regimes where solitons exist as stable/unstable structures. We present eight classes of dissipative solitons from which six are known (stationary, pulsating, vortex spinning, filament, exploding, creeping) and two are novel (creeping-vortex propellers and spinning "bean-shaped" solitons). By running lengthy simulations for the different parameters of the equation, we find ranges of existence of stable structures (stationary, pulsating, circular vortex spinning, organized exploding), and unstable structures (elliptic vortex spinning that leads to filament, disorganized exploding, creeping). Moreover, by varying even the two initial conditions together with vorticity, we find a richer behavior in the form of creeping-vortex propellers, and spinning "bean-shaped" solitons. Each class differentiates from the other by distinctive features of their energy evolution, shape of initial conditions, as well as domain of existence of parameters.

Concept erasure methods that operate via linear projection assume that features occupy separable subspaces. We show this assumption fails under dense superposition: when two features are forced into an antipodal pair sharing a single subspace, state-of-the-art linear erasure destroys both, not just the target. Networks trained with gradient descent instead solve this problem non-linearly, but not uniformly: they converge to one of two distinct circuit-level solutions depending on initialization, which we call mirror and shadow solutions. We map this bifurcation as a function of feature entanglement, show it reflects a stable attractor structure rather than an artifact of our setup, and use targeted causal interventions to demonstrate that both solutions leave a substantial, measurable trace of the erased feature's representation intact, recoverable through a single scalar patch rather than requiring any further training. This mirrors a failure mode recently observed empirically in LLM unlearning, where suppression rather than deletion allows forgotten knowledge to resurface; our results offer a mechanistic, causally-validated account of why that failure mode occurs.

Two-dimensional Kolmogorov flow is a common proving ground for the dynamical systems picture of turbulence. Hundreds of unstable periodic orbits have been found for this system across many Reynolds numbers and domain sizes, although the most well explored is Re=40 on a square 2πtimes 2π domain with a forcing wavenumber of k=4. Here I present a catalog of 8952 exact steady states of the flow in this canonical parameter regime. These solutions provide a fixed constellation informing the dynamics of both turbulence and transients.

We study the dynamics of Stochastic Gradient Descent (SGD), which is known to steer deep neural networks toward invariant sets that correspond to simpler subnetworks. How this steering unfolds over time remains poorly understood. We answer this by modeling the stochastic gradient flow (SGF) as a percolation process, in which architectural symmetries force subnetworks to merge in discrete simultaneous blocks rather than one at a time. These structural transitions register as variance spikes in a macroscopic order parameter, echoing physical phase transitions. We further show this trapping mechanism and its associated scaling cascade extend to Adam and AdamW under an explicit heavy-tailed noise model.

Chaos and unpredictability are traditionally synonymous, yet large-scale machine learning methods recently have demonstrated a surprising ability to forecast chaotic systems well beyond typical predictability horizons. However, recent works disagree on whether specialized methods grounded in dynamical systems theory, such as reservoir computers or neural ordinary differential equations, outperform general-purpose large-scale learning methods such as transformers or recurrent neural networks. These prior studies perform comparisons on few individually-chosen chaotic systems, thereby precluding robust quantification of how statistical modeling choices and dynamical invariants of different chaotic systems jointly determine empirical predictability. Here, we perform the largest to-date comparative study of forecasting methods on the classical problem of forecasting chaos: we benchmark 24 state-of-the-art forecasting methods on a crowdsourced database of 135 low-dimensional systems with 17 forecast metrics. We find that large-scale, domain-agnostic forecasting methods consistently produce predictions that remain accurate up to two dozen Lyapunov times, thereby accessing a new long-horizon forecasting regime well beyond classical methods. We find that, in this regime, accuracy decorrelates with classical invariant measures of predictability like the Lyapunov exponent. However, in data-limited settings outside the long-horizon regime, we find that physics-based hybrid methods retain a comparative advantage due to their strong inductive biases.

We investigate diffusion approximations for a class of multivariate inhomogeneous finite-velocity random motions motivated by models of active particles. A particle alternates cyclically among prescribed velocity fields V_1,dots,V_p with random run times and evolves according to either a flight dynamics, consisting of piecewise-linear motion, or a gliding dynamics, in which the particle follows the corresponding velocity flow. Under a Kac-type scaling that couples vanishing run times with diverging particle speed, we prove weak convergence of both processes to multidimensional diffusions. It turns out that the order of successive cyclic motions affects the diffusion limit. In addition to the second-order term generated by fluctuations of the random run times, the limiting drift contains directional derivatives DV_i[V_j] of the underlying vector fields. In Stratonovich form, part of this drift is expressed through their Lie brackets [V_i,V_j]. Thus, the generic non-commutativity of the microscopic motions survives the diffusive scaling and generates a macroscopic drift which depends on the cyclic order. The limiting drift also distinguishes the flight and gliding dynamics, despite their being driven by the same vector fields and run times. Several examples, including run-and-reverse motion and cyclic dynamics generated by multiple vector fields, illustrate how the cyclic switching protocol and the geometry of the underlying velocity fields shape the limiting diffusions.

Exact firing rate models, also known as next-generation neural mass models (NG-NMMs), provide a rigorous description of the dynamics of neural populations. While in its simplest form a single population only displays fixed-point activity, multi-population models may display a range of different behaviors. In this work, we study the dynamics of all-excitatory or all-inhibitory NG-NMMs coupled through sparse random networks with row-normalized network topology. Linear stability analysis of the homogeneous states of the system, representing asynchronous neural activity, provides a dispersion relation linking the emergence of spatiotemporal dynamics to the spectra of the connectivity matrix. Using bounds from random matrix theory, we identify the parameter regions where instabilities occur. In undirected networks, only inhibitory systems produce heterogeneous stationary patterns, corresponding to a winner-takes-all mechanism. In directed networks, exotic rhythmic states with high frequencies emerge in both, excitatory and inhibitory systems. Numerical simulations reveal that these hectic oscillatory states correspond to high-dimensional chaos with extensive properties.

We study the chemical distance of supercritical Bernoulli percolation on Z^d. Recently, Dembin [Dem22] showed that the chemical distance exhibits sublinear variance, a phenomenon now referred to as superconcentration. In this article, we establish an equivalence between this phenomenon and chaotic behavior of geodesics under small perturbations of the configuration, thereby confirming Chatterjee's general principle relating anomalous fluctuations to chaos in the context of Bernoulli percolation. Our methods rely on a dynamical version of the effective radius, refining the notion first proposed in [CN25], in order to measure the co-influence of a given edge whose weight may be infinite. Together with techniques from the theory of lattice animals, this approach allows us to quantify the total co-influence of edges in terms of the overlap between original and perturbed geodesics.

A scuba diver holding constant depth balances on an unstable equilibrium: the gas carried in the suit and buoyancy compensator compresses with depth, so the buoyant force falls as the diver sinks and rises as the diver ascends. We represent the diver as a proportional-derivative controller that regulates this compressible-buoyancy saddle after a finite reaction delay, and we derive the governing delay differential equation from the vertical force balance and the isothermal gas law, reducing it to a damping ratio, two control gains, and a dimensionless delay. The characteristic spectrum, obtained by pseudospectral collocation of the semigroup generator and checked against a direct Newton solution of the characteristic equation, locates the Hopf boundary that separates stable hovering from sustained porpoising; for the baseline diver the critical reaction delay is 3.36 s and the onset period is 28.8 s. The bifurcation is supercritical, and because the saturating force is the quadratic hydrodynamic drag, the limit-cycle amplitude grows in proportion to the delay excess rather than as its square root. The safe-operating envelope shows that runaway ascent is triggered by saturation of the compensator, not by loss of linear stability, so a stable and an unstable diver can share the same escape threshold. As onset is approached, the lag-one autocorrelation and variance rise while the fitted recovery rate falls and matches the spectral abscissa, giving an eigenvalue-exact early warning of the transition.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

Synchronization in one dimension displays generic scale invariance with universal properties previously observed in surface kinetic roughening and the wider context of the Kardar-Parisi-Zhang (KPZ) universality class. This has been established for phase oscillators and also for some limit-cycle oscillators, both in the presence of columnar (quenched) disorder and of time-dependent noise, by extensive numerical simulations, and has been analytically motivated by continuum approximations in the strong oscillator coupling limit. The robustness and the precise boundaries in parameter space for such critical behavior remain unclear, however, which may preclude further developments, including the extension of these results to higher dimensions and the experimental observation of nonequilibrium criticality in synchronizing (e.g.~electronic or chemical) oscillators. We here present complete numerical phase diagrams of one-dimensional synchronization, including saturation times and values, but, most importantly, also dynamical features giving insight into the gradual emergence of synchronous dynamics, based on systems of phase oscillators with either type of randomness. In the absence of synchronization, the dynamics evolves as expected for random deposition (for time-dependent noise) or linear growth (for columnar disorder), while a crossover from Edwards-Wilkinson to Kardar-Parisi-Zhang behavior (with the corresponding type of randomness) is observed as the randomness strength, or the nonoddity of the coupling among oscillators, is increased in the synchronous region -- their combined effect being partially captured by the so-called KPZ coupling. The distortion of scaling due to phase slips near the desynchronization boundary, a feature that is likely to play a role in experimental contexts, is also discussed.

Weather Jiu-Jitsu is a control paradigm for extreme climatological events, inspired by chaos theory. As a proposition, small, precise, targeted, and cost-inexpensive perturbations can redirect trajectories of a large dynamical system. This strategy has been demonstrated analytically in the Lorenz-63 system, where a naturally chaotic trajectory switching between two attractors can be confined to a single attractor, indefinitely, via arbitrarily small perturbations. This paper examines the feasibility of Microsoft's Aurora -- a 1.3 billion parameter global atmospheric model -- as a test bed for this strategy. This paper explores three questions: (1) Is Aurora a reliable enough simulation environment to serve as a meaningful testbed? (2) Are the perturbations required to redirect its trajectories small enough to be physically plausible? (3) Does Aurora's learned latent space (the parametric estimators on climatological attributes) yield any apparent, structured, and/or perhaps interpretable features that can convey a geo/atmospheric response to initial conditions? We find evidence consistent with all three: Aurora's modeled trajectories respond to perturbations beyond measurement drift, the perturbation magnitudes required are small relative to the model's own forecast uncertainty, and its latent representations exhibit directional structure that responds to Jiu-Jitsu-type interventions, even though that structure does not separate extreme from normal states outright. These results should be read as feasibility diagnostics rather than a demonstration of control: we do not implement or test an actual steering intervention on Aurora, and several of our findings, particularly around the model's latent-space geometry, are exploratory.

We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and the grokking time distributions and compare them with the prediction of the proposed grokking model. Finally, we numerically analyse the connection between structure formation and grokking.

The emergence and impact of tipping points have garnered significant interest in both the social and natural sciences. Despite widespread recognition of the importance of feedbacks between human and natural systems, it is often assumed that the observed nonlinear dynamics in these coupled systems rests within either underlying human or natural processes, rather than the rates at which they interact. Using adoption of agricultural diversification practices as a case study, we show how two stable management paradigms (one dominated by conventional, homogeneous practices, the other by diversified practices) can emerge purely from temporal feedbacks between human decisions and ecological responses. We explore how this temporal mechanism of tipping points provides insight into designing more effective interventions that promote farmers transitions towards sustainable agriculture. Moreover, our flexible modeling framework could be applied to other cases to provide insight into numerous questions in social-ecological systems research and environmental policy.

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

Legged locomotion is a challenging task in the field of robotics but a rather simple one in nature. This motivates the use of biological methodologies as solutions to this problem. Central pattern generators are neural networks that are thought to be responsible for locomotion in humans and some animal species. As for robotics, many attempts were made to reproduce such systems and use them for a similar goal. One interesting design model is based on spiking neural networks. This model is the main focus of this work, as its contribution is not limited to engineering but also applicable to neuroscience. This paper introduces a new general framework for building central pattern generators that are task-independent, biologically plausible, and rely on learning methods. The abilities and properties of the presented approach are not only evaluated in simulation but also in a robotic experiment. The results are very promising as the used robot was able to perform stable walking at different speeds and to change speed within the same gait cycle.

We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts.

Autoregressive models are trained to predict a system's behavior one step at a time, and recursive generation allows the learned dynamics to unfold over long horizons. To what extent can such dynamics learned from local observations recover broader organization of an underlying system that was only partially observed during training? Here we study small autoregressive transformers trained from scratch on trajectories sampled from restricted parameter regimes of several non-linear dynamical systems, including logistic and sine maps, the Lorenz system, and the generalized Hopf system, with control parameters and state trajectories represented as sequences of continuous tokens. Under closed-loop evaluation at parameters far outside the training distribution, the models can recover self-similar period-doubling cascades, chaotic dynamics, and attractor structures with remarkable visual and numerical fidelity. For the logistic map, a transformer reproduces successive period doublings up to period 128, yielding a finite-order scaling ratio of 4.6687, matching the Feigenbaum constant to within 5times10^{-4}. We further investigate how these structures emerge over the course of training, and reveal with causal interventions how control-parameter information is processed through attention into state prediction and shapes the resulting closed-loop dynamics. These results suggest that a surprisingly narrow window into a system's local behavior may suffice for autoregressive transformers to generalize to its unseen global dynamical organization.

We obtain closed-form solutions of several inhomogeneous Lienard equations by the factorization method. The two factorization conditions involved in the method are turned into a system of first-order differential equations containing the forcing term. In this way, one can find the forcing terms that lead to integrable cases. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. The illustrative examples of Lienard solutions obtained in this way generically have rational parts, and consequently display singularities.

We study the scaling properties of avalanche activity in the two-dimensional Abelian sandpile model. Instead of the conventional avalanche size distribution, we analyze the site activity distribution, which measures how often a site participates in avalanches when grains are added across the lattice. Using numerical simulations for system sizes up to \(L = 160\), averaged over \(10^4\) configurations, we determine the probability distribution \(P(A, L)\) of site activities. The results show that \(P(A, L)\) follows a finite-size scaling form \[ P(A, L) \sim L^{-2} F\Big(A{L^2}\Big). \] For small values \(A \ll L^2\) the scaling function behaves as \[ F(u) \sim u^{-1/2}, \quad corresponding to \quad P(A) \sim 1{L}, \] while for large activities \(A \sim O(L^2)\) the distribution decays as \[ F(u) \sim \exp\big(-c_3 u - c_4 u^2\big). \] The crossover between these two regimes occurs at \[ A^* \sim 0.1 \, L^2, \] marking the threshold between typical and highly excitable sites. This characterization of local avalanche activity provides complementary information to the usual avalanche size statistics, highlighting how local regions serve as frequent conduits for critical dynamics. These results may help connect sandpile models to real-world self-organized critical systems where only partial local activity can be observed.

Outer-learning algorithms use infinitesimal sensitivities to propose finite changes to initialization or training parameters. For hard-ReLU training, the derivative of the finite program and the derivative of its flow limit do not by themselves specify the response at a chosen radius. We characterize the intervening regime in which the perturbation radius is proportional to the GD step. Integer event rounding then survives at leading order: smooth Euler bias shifts each discrete phase, and upstream rounding moves downstream branch boundaries. We derive the crossing indices and a uniform endpoint expansion for finitely many separated transverse events in piecewise-C^2 dynamics, away from recursive phase boundaries. In contractive affine regions, an explicit remainder and complete branch verification certify finite candidate comparisons. Scalar phase frequencies and a coupled feedback ablation test the mechanism; frozen nonlinear-network experiments show radius-dependent prediction accuracy, including incomplete branch matches and failed-word tails. Together with local AD and uniform flow consistency, the result identifies sufficient response regimes: differentiating training is a choice of perturbation resolution as well as a choice of derivative.

In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.

Let (M omega) be a two dimensional symplectic manifold, phi: M to M a symplectomorphism with hyperbolic fixed point x and transversely intersecting stable and unstable manifolds W^s(phi, x) cap W^u(phi, x)=:H(phi, x). The intersection points are called homoclinic points, and the stable and unstable manifold are in this situation Lagrangian submanifolds. For this Lagrangian intersection problem with its infinite number of intersection points and wild oscillation behavior, we first define a Floer homology generated by finite sets of so-called contractible homoclinic points. This generalizes very significantly the Floer homologies generated by (semi)primary points defined by us in earlier works. Nevertheless these Floer homologies only consider quite `local' aspects of W^s(phi, x) cap W^u(phi, x) since their generator sets are finite, but the number of all contractible homoclinic points is infinite. To overcome this issue, we construct a direct limit of these `local' homoclinic Floer homologies over suitable index sets. These direct limits thus accumulate the information gathered by the finitely generated local' homoclinic Floer homologies.

We study spontaneous scalarization of static, spherically symmetric black holes in scalar--Gauss--Bonnet gravity with a Ricci coupling F=1 βϕ^{2}/4+γβ^{2}ϕ^{4}+O(ϕ^{6}) that nonlinearly completes the quadratic one. For arbitrary couplings and potential, we derive the near-horizon regularity condition and horizon curvature invariants in closed form; scalarized branches terminate where this condition degenerates, at finite horizon curvature. A weakly nonlinear expansion fixes the bifurcation direction exactly: for a massless scalar, its governing coefficient is quadratic in β, changes sign at β=1.1156, and depends linearly on a quartic self-interaction. At equal horizon values of the scalar, completions differ in mass, to leading order, in proportion to γ. A radial-perturbation master equation for arbitrary couplings confirms the exchange of stability at the bifurcation and turning points. Consequently, the domain of radially stable black holes shrinks as γ increases over the range studied; it is smaller for the exponential completion than for the quadratic truncation.

Water looks unchanged as it warms, then at a critical point it boils. We ask whether long-horizon language agents show an analogous transition in their implicit world models. In some parameter settings, changing state load by a small amount, or adding a single step of horizon, leaves behavior nearly unchanged; near a critical boundary, the same small change causes a sudden world collapse. We study this effect in a deterministic task family with exact per-step gold state. A large grid search over state cardinality, dependency density, horizon, branching, observation mode, and mutation rate reveals a phase diagram: a solved plateau, a narrow transition band, and a collapse floor. Per-step traces show the mechanism: world-state fidelity fails before action validity, so the agent is not merely choosing a bad action; it is acting from a corrupted world. Stronger models translate the critical boundary but do not remove the qualitative transition. These results make world-model collapse a measurable bottleneck for long-horizon agents.

In heterogeneous networks of coupled oscillators, phase frustration typically prevents the emergence of synchronization in the Sakaguchi--Kuramoto (SK) model. In this study, we propose an analytical framework to overcome this barrier and induce complete synchronization at a specified small coupling value in oscillators governed by phase-frustrated bi-harmonic coupling. We derive an optimal set of natural frequencies that is robust against added noise and correlated with the network degree heterogeneity, along with the parameters involved in the bi-harmonic coupling function that lead to complete synchronization (r = 1). In addition, we find complete synchronization transitions accompanied by hysteresis in scale-free networks, indicating a first-order (discontinuous) phase transition, whereas Erdős--Rényi networks exhibit complete synchronization through a second-order (continuous) phase transition. Furthermore, we use the mean-field approximation in the presence of optimal frequencies to determine the critical coupling strength associated with the synchronization transition in the pure second-harmonic Sakaguchi--Kuramoto model. Here, the obtained optimal natural frequencies scale linearly with the node degree, and the critical coupling strength for the onset of synchronization is derived analytically from the self-consistent equations. In this specific regime, we observe a perfectly ordered two-cluster synchronized state. These findings remain robust for higher-order harmonic coupling schemes, as well as across a diverse range of synthetic and empirical networks, including scale-free, Erdős--Rényi, Zachary Karate Club, and the C.~elegans neural network, demonstrating their general applicability.

We apply a Gaussian process method to the extreme gamma-ray flares of 3C 454.3 and 3C 279 to discover the variable patterns and then to investigate the physical origins of the giant flares. The kernels of stochastically driven damped simple harmonic oscillator (SHO), the damped random-walk (DRW), and Matrm ern-3/2 are respectively used to describe the adaptive-binning gamma-ray light curves of the two flares. Our findings show that both the extreme gamma-ray flares of 3C 454.3 and 3C 279 clearly prefer the SHO kernel in the over-damped mode and the Matrm ern-3/2 kernel over the DRW kernel. The resulted SHO and Matrm ern-3/2 power spectral densities (PSDs) are the same for each object, with the index changing from -4 at high frequencies to 0 at low frequencies. The patterns of the two flares are both approaching the critical damping mode with the quality factor Q approx 0.4 (i.e., the damping ratio eta approx 1.25), but with slightly different damping timescales. The characteristic timescale (corresponding to the broken frequency in the PSD) for 3C 454.3 is 2-3 days and 3-5 days for 3C 279. The variable patterns found here suggest that once the system responds to the energy injection disturbance, the release of the energy in the system is finished abruptly. The obtained timescale provides a constraint on the size of energy dissipation region for each source.

The symbolic discovery of Ordinary Differential Equations (ODEs) from trajectory data plays a pivotal role in AI-driven scientific discovery. Existing symbolic methods predominantly rely on fixed, pre-collected training datasets, which often result in suboptimal performance, as demonstrated in our case study in Figure 1. Drawing inspiration from active learning, we investigate strategies to query informative trajectory data that can enhance the evaluation of predicted ODEs. However, the butterfly effect in dynamical systems reveals that small variations in initial conditions can lead to drastically different trajectories, necessitating the storage of vast quantities of trajectory data using conventional active learning. To address this, we introduce Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching (APPS). Instead of directly selecting individual initial conditions, our APPS first identifies an informative region within the phase space and then samples a batch of initial conditions from this region. Compared to traditional active learning methods, APPS mitigates the gap of maintaining a large amount of data. Extensive experiments demonstrate that APPS consistently discovers more accurate ODE expressions than baseline methods using passively collected datasets.

Traveling waves of activity in neural tissue can be halted by spatial inhomogeneity in synaptic coupling. We study an integrate-and-fire network in which each neuron fires once and the coupling decays exponentially. For this model the leading-edge firing map reduces exactly to a scalar equation for the wave speed in space, with a slow unstable and a fast stable homogeneous speed, c_1 and c_2, and a periodic modulation of the coupling enters this equation pointwise. Positive periodic waves terminate in folds. For slowly varying modulation the fold amplitude approaches a plateau set by the trough of the modulation, where the local bottleneck speed is c_1 c_2; the approach to the plateau is set by the local geometry of the trough. For rapidly varying modulation the fold amplitude grows linearly with frequency, with a slope given by a full-amplitude average over the integrated modulation profile; a weak-ripple truncation of that average overestimates the slope about 2.5-fold at the default coupling. A perturbative recursion in the amplitude gives the speed profile explicitly. We distinguish loss of the periodic wave from failure of a particular launch, and the excitatory regime εle 1 from its sign-changing extension. The results are checked against direct integration of the reduced equation and first-spike firing-map simulations of the network.

The generation of sustained, open-ended complexity from local interactions remains a fundamental challenge in artificial life. Differentiable multi-agent systems, such as Petri Dish Neural Cellular Automata (PD-NCA), exhibit rich self-organization driven purely by spatial competition; however, they are highly sensitive to hyperparameters and frequently collapse into uninteresting patterns and dynamics, such as frozen equilibria or structureless noise. In this paper, we introduce PBT-NCA, a meta-evolutionary algorithm that evolves a population of PD-NCAs subject to a composite objective that rewards both historical behavioral novelty and contemporary visual diversity. Driven by this continuous evolutionary pressure, PBT-NCA spontaneously generates a plethora of emergent lifelike phenomena over extended horizons-a hallmark of true open-endedness. Strikingly, the substrate autonomously discovers diverse morphological survival and self-organization strategies. We observe highly regular, coordinated periodic waves; spore-like scattering where homogeneous groups eject cell-like clusters to colonize distant territories; and fluid, shape-shifting macro-structures that migrate across the substrate, maintaining stable outer boundaries that enclose highly active interiors. By actively penalizing monocultures and dead states, PBT-NCA sustains a state of effective complexity that is neither globally ordered nor globally random, operating persistently at the "edge of chaos".

We show that the Turing complete rule 110 in elementary cellular automata can be emulated by alternating three simple symmetric rules in space. As the constituent rules themselves are trivial, this shows that diversity alone can generate emergent phenomena. The rules cannot distinguish left from right, and the chirality of rule 110 is created by their arrangement. Random initial conditions lock on to rule 110 over 90% of the time, explained through the motion of domain walls between desynchronized regions. Adding random noise creates errors that seed domain walls, whose fraction scales with the square root of noise strength, explained through a simple rate equation. This points to a route to complexity through repeated, perturbed units, common in biology, and shows that in cellular automaton models of quantum field theory, broken symmetries such as parity need not be present in the local rules.

We propose an algorithm for change point monitoring in linear causal models that accounts for interventions. Through a special centralization technique, we can concentrate the changes arising from causal propagation across nodes into a single dimension. Additionally, by selecting appropriate intervention nodes based on Kullback-Leibler divergence, we can amplify the change magnitude. We also present an algorithm for selecting the intervention values, which aids in the identification of the most effective intervention nodes. Two monitoring methods are proposed, each with an adaptive intervention policy to make a balance between exploration and exploitation. We theoretically demonstrate the first-order optimality of the proposed methods and validate their properties using simulation datasets and two real-world case studies.

Continuous cellular automata are rocketing in popularity, yet developing a theoretical understanding of their behaviour remains a challenge. In the case of Lenia, a few fundamental open problems include determining what exactly constitutes a soliton, what is the overall structure of the parameter space, and where do the solitons occur in it. In this abstract, we present a new method to automatically classify Lenia systems into four qualitatively different dynamical classes. This allows us to detect moving solitons, and to provide an interactive visualization of Lenia's parameter space structure on our website https://lenia-explorer.vercel.app/. The results shed new light on the above-mentioned questions and lead to several observations: the existence of new soliton families for parameters where they were not believed to exist, or the universality of the phase space structure across various kernels.

In order to study oscillators coupled by pulses emitted at discrete times, it is generally assumed that the effect of each pulse is an instantaneous phase shift, with a possible conduction delay between emission and receipt. However, we are interested in neural oscillators in which each pulse (action potential) activates a biexponential synapse, and at high frequencies, the duration of the synaptic conductance may be greater than the network period. Therefore, the effect of received pulses can summate over the course of several cycles. To overcome this limitation, we previously defined a mean field approach for a single synchronous population. Here, we extend it to two alternating synchronous populations by assuming a perturbation of a single oscillator from one of the synchronous populations which is represented by a single self-connected neural oscillator. The train of delayed biexponential synapses emitted by each oscillator is divided into a tonic and a phasic component. Because of the continuous rather than pulsatile nature of the input, we had to drop the concept of oscillator phase and derive self-consistent criteria for the existence and stability of an alternating firing pattern between two synchronous populations based solely on the perturbation of time intervals from their steady state values. Simulations showed that for high frequency oscillations as defined above, the mean field approach predicts the existence and stability of phase-locking between two synchronous populations better than the instantaneous phase shift approach. This approach may generalize to other forms of coupling amongst non-neural oscillators.

We study the Lane--Emden equation of index n=5, the critical polytropic case separating finite-radius from infinite-radius self-gravitating equilibria. The scaling family θ_λ(ξ)=λ^{1/2}θ(λξ) generates a neutral mode ϕ_0=tfrac12θ+ξθ' about the Schuster profile. Schuster's transformation and a logarithmic variable reduce the equation to the autonomous Hamiltonian system ddotν=tfrac14(ν-ν^5) with a first integral. Besides the Schuster (separatrix) and Srivastava solutions, every orbit crossing ν=0 is given in closed form by a Weierstrass wp-function, and the closed forms agree with direct numerical integration to about 10^{-12}. Up to scaling and sign, the Schuster solution is the only solution regular at the center, and it separates positive solutions from solutions with infinitely many zeros. In homology-invariant variables the singular solution θ=(2ξ)^{-1/2} is a center exactly at n=5, where its stability changes. For a distance-varying Emden--Fowler system with a shifted equilibrium we prove that at n=5 the unique equilibrium is linearly and nonlinearly unstable. Finally, the linearization about the Schuster profile is the Schrödinger operator L=-Δ_r-5θ_S^4 on L^2((0,infty),ξ^2dξ). Its Friedrichs extension has essential spectrum [0,infty), a threshold resonance at 0 carried by the non-normalizable scaling mode, and exactly one negative eigenvalue, λ_0approx-1.21, so the Schuster profile is a saddle point of the associated energy functional.

Permutation Entropy and statistiCal Complexity Analysis for astRophYsics (PECCARY) is a computationally inexpensive, statistical method by which any time-series can be characterized as predominantly regular, complex, or stochastic. Elements of the PECCARY method have been used in a variety of physical, biological, economic, and mathematical scenarios, but have not yet gained traction in the astrophysical community. This study introduces the PECCARY technique with the specific aims to motivate its use in and optimize it for the analysis of astrophysical orbital systems. PECCARY works by decomposing a time-dependent measure, such as the x-coordinate or orbital angular momentum time-series, into ordinal patterns. Due to its unique approach and statistical nature, PECCARY is well-suited for detecting preferred and forbidden patterns (a signature of chaos), even when the chaotic behavior is short-lived or when working with a relatively short duration time-series or small sets of time-series data. A variety of examples are used to demonstrate the capabilities of PECCARY. These include mathematical examples (sine waves, varieties of noise, sums of sine waves, well-known chaotic functions), a double pendulum system, and astrophysical tracer particle simulations with potentials of varying intricacies. Since the adopted timescale used to diagnose a given time-series can affect the outcome, a method is presented to identify an ideal sampling scheme, constrained by the overall duration and the natural timescale of the system. The accompanying PECCARY Python package and its usage are discussed.

Recurrent neural networks (RNNs) provide a powerful approach in neuroscience to infer latent dynamics in neural populations and to generate hypotheses about the neural computations underlying behavior. However, past work has focused on relatively simple, input-driven, and largely deterministic behaviors - little is known about the mechanisms that would allow RNNs to generate the richer, spontaneous, and potentially stochastic behaviors observed in natural settings. Modeling with Hidden Markov Models (HMMs) has revealed a segmentation of natural behaviors into discrete latent states with stochastic transitions between them, a type of dynamics that may appear at odds with the continuous state spaces implemented by RNNs. Here we first show that RNNs can replicate HMM emission statistics and then reverse-engineer the trained networks to uncover the mechanisms they implement. In the absence of inputs, the activity of trained RNNs collapses towards a single fixed point. When driven by stochastic input, trajectories instead exhibit noise-sustained dynamics along closed orbits. Rotation along these orbits modulates the emission probabilities and is governed by transitions between regions of slow, noise-driven dynamics connected by fast, deterministic transitions. The trained RNNs develop highly structured connectivity, with a small set of "kick neurons" initiating transitions between these regions. This mechanism emerges during training as the network shifts into a regime of stochastic resonance, enabling it to perform probabilistic computations. Analyses across multiple HMM architectures - fully connected, cyclic, and linear-chain - reveal that this solution generalizes through the modular reuse of the same dynamical motif, suggesting a compositional principle by which RNNs can emulate complex discrete latent dynamics.

We prove a persistence theorem for attracting weakly normally hyperbolic invariant tori under small time-periodic perturbations. The theorem extends recent continuation results for weakly normally hyperbolic limit cycles to invariant tori of arbitrary dimension, providing a general analytical framework for their detection. As an application, we establish an averaging principle showing that attracting normally hyperbolic invariant d-tori of the averaged system give rise to attracting normally hyperbolic invariant (d+1)-tori of the original non-autonomous system. We further introduce a polynomiality-preserving construction that simultaneously lifts the dimensions of the phase space and the attracting normally hyperbolic invariant tori, yielding recursive lower bounds for the maximal number of codimension-1 normally hyperbolic invariant tori of polynomial vector fields of a given degree, thereby extending the counting aspect of Hilbert's sixteenth problem to higher-dimensional invariant tori. In particular, we prove that this number grows at least polynomially with the degree and becomes unbounded for invariant tori of higher codimension.

We address hallucination detection in Large Language Models (LLMs) and Multimodal Large Language Models (MLLMs) by framing the problem through the lens of dynamical systems stability theory. Rather than treating hallucination as a straightforward classification task, we conceptualize (M)LLMs as dynamical systems, where factual knowledge is represented by stable equilibrium points within the representation space. Our main insight is that hallucinations tend to arise at the boundaries of knowledge-transition regions separating stable and unstable zones. To capture this phenomenon, we propose Lyapunov Probes: lightweight networks trained with derivative-based stability constraints that enforce a monotonic decay in confidence under input perturbations. By performing systematic perturbation analysis and applying a two-stage training process, these probes reliably distinguish between stable factual regions and unstable, hallucination-prone regions. Experiments on diverse datasets and models demonstrate consistent improvements over existing baselines.

This chapter offers a principled approach to the prediction of chaotic systems from data. First, we introduce some concepts from dynamical systems' theory and chaos theory. Second, we introduce machine learning approaches for time-forecasting chaotic dynamics, such as echo state networks and long-short-term memory networks, whilst keeping a dynamical systems' perspective. Third, the lecture contains informal interpretations and pedagogical examples with prototypical chaotic systems (e.g., the Lorenz system), which elucidate the theory. The chapter is complemented by coding tutorials (online) at https://github.com/MagriLab/Tutorials.

We revisit Tirole's classic paper "Asset Bubbles and Overlapping Generations" (1985, Econometrica) in the case of a dividend-paying asset. Recently, Pham and Toda (2026) constructed a counterexample to Proposition 1(c), showing that Tirole's equilibrium classification is incorrect as stated and that long-run outcomes can depend on initial capital. This paper characterizes the entire set of equilibrium initial asset prices under capital over-accumulation. Exactly one of three regimes occurs: (i) a unique bubbleless equilibrium with capital converging to zero (capital collapse), (ii) a unique asymptotically bubbly equilibrium converging to a positive steady state (bubble necessity), or (iii) a continuum of equilibria with different long-run bubble behavior (indeterminacy). We further derive a threshold for initial capital under the bubble necessity condition, establish preference-free sufficient conditions for capital collapse, and show that the continuum in the pure bubble model survives sufficiently small dividend perturbations. Closed-form examples illustrate the possible long-run outcomes.

We introduce spatiotemporal solitons of the two-dimensional complex Ginzburg-Landau equation (2D CCQGLE) with cubic and quintic nonlinearities in which asymmetry between space-time variables is included. The 2D CCQGLE is solved by a powerful Fourier spectral method, i.e., a Fourier spatial discretization and an explicit scheme for time differencing. Varying the system's parameters, and using different initial conditions, numerical simulations reveal 2D solitons in the form of stationary, pulsating and exploding solitons which possess very distinctive properties. For certain regions of parameters, we have also found stable coherent structures in the form of spinning (vortex) solitons which exist as a result of a competition between focusing nonlinearities and spreading while propagating through medium.

The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden-Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden-Fowler equation are discussed in the same context

We present a model for chi^{(2)} waveguides accounting for three modes, two of which make an avoided crossing at the second harmonic wavelength. We introduce two linearly coupled pure modes and adjust the coupling to replicate the waveguide dispersion near the avoided crossing. Analysis of the nonlinear system reveals continuous wave (CW) solutions across much of the parameter-space and prevalence of its modulational instability. We also predict the existence of the avoided-crossing solitons, and study peculiarities of their dynamics and spectral properties, which include formation of a pedestal in the pulse tails and associated pronounced spectral peaks. Mapping these solitons onto the linear dispersion diagrams, we make connections between their existence and CW existence and stability. We also simulate the two-color soliton generation from a single frequency pump pulse to back up its formation and stability properties.

The presence of linear paths in parameter space between two different network solutions in certain cases, i.e., linear mode connectivity (LMC), has garnered interest from both theoretical and practical fronts. There has been significant research that either practically designs algorithms catered for connecting networks by adjusting for the permutation symmetries as well as some others that more theoretically construct paths through which networks can be connected. Yet, the core reasons for the occurrence of LMC, when in fact it does occur, in the highly non-convex loss landscapes of neural networks are far from clear. In this work, we take a step towards understanding it by providing a model of how the loss landscape needs to behave topographically for LMC (or the lack thereof) to manifest. Concretely, we present a `mountainside and ridge' perspective that helps to neatly tie together different geometric features that can be spotted in the loss landscape along the training runs. We also complement this perspective by providing a theoretical analysis of the barrier height, for which we provide empirical support, and which additionally extends as a faithful predictor of layer-wise LMC. We close with a toy example that provides further intuition on how barriers arise in the first place, all in all, showcasing the larger aim of the work -- to provide a working model of the landscape and its topography for the occurrence of LMC.

Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we are interested in reconstructing such systems from observed time series for prediction or mechanistic insight, where by reconstruction we mean learning geometrical and invariant temporal properties of the system in question (like attractors). However, training reconstruction algorithms like recurrent neural networks (RNNs) on such systems by gradient-descent based techniques faces severe challenges. This is mainly due to exploding gradients caused by the exponential divergence of trajectories in chaotic systems. Moreover, for (scientific) interpretability we wish to have as low dimensional reconstructions as possible, preferably in a model which is mathematically tractable. Here we report that a surprisingly simple modification of teacher forcing leads to provably strictly all-time bounded gradients in training on chaotic systems, and, when paired with a simple architectural rearrangement of a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful reconstruction in spaces of at most the dimensionality of the observed system. We show on several DS that with these amendments we can reconstruct DS better than current SOTA algorithms, in much lower dimensions. Performance differences were particularly compelling on real world data with which most other methods severely struggled. This work thus led to a simple yet powerful DS reconstruction algorithm which is highly interpretable at the same time.

In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets.

To model high-risk traffic densities a cellular automaton model is constructed, exhibiting Ising model-like properties. The attraction-repulsion forces between vehicles are evaluated as coupling, and the coupling function p(1-p)=g/8, where p is the initial distribution of cells with state 1, and g is the Moore neighbor count, which has roots cos^2(π/8) and sin^2(π/8) for g=1. This is achieved without the use of any trigonometric functions in the code. From the roots, the tangent polynomial tan^2(x) + tan(x) emerges. Kramers-Wannier duality is recovered and it is conjectured that the 1/2 difference between the roots serves as a fixed point for a projection mechanism from the 2-dimensional Ising model onto a 1-dimensional Ising chain through the Gudermannian function. The sigmoid evaluated at the proposed fixed point is substituted into the derivative of the logistic coupling function, σ(1/2)(1-σ(1/2)), yielding a numerical approximation to the three-dimensional Ising inverse critical coupling. Finally, the results are linked to risk densities in traffic and vehicle types, accounting for the amplification of fatal accidents.

Attractor dynamics are a fundamental computational motif in neural circuits, supporting diverse cognitive functions through stable, self-sustaining patterns of neural activity. In these lecture notes, we review four key examples that demonstrate how autoassociative neural network models can elucidate the computational mechanisms underlying attractor-based information processing in biological neural systems performing cognitive functions. Drawing on empirical evidence, we explore hippocampal spatial representations, visual classification in the inferotemporal cortex, perceptual adaptation and priming, and working-memory biases shaped by sensory history. Across these domains, attractor network models reveal common computational principles and provide analytical insights into how experience shapes neural activity and behavior. Our synthesis underscores the value of attractor models as powerful tools for probing the neural basis of cognition and behavior.

Singular limits for the following indirect signalling chemotaxis system align* \left\{ array{lllllll} \partial_t n = \Delta n - \nabla \cdot (n \nabla c ) & in \Omega\times(0,\infty) , \varepsilon \partial_t c = \Delta c - c + w & in \Omega\times(0,\infty), \varepsilon \partial_t w = \tau \Delta w - w + n & in \Omega\times (0,\infty), \partial_\nu n = \partial_\nu c = \partial_\nu w = 0, &on \partial\Omega\times (0,\infty) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & on \Omega, array \right. align* are investigated. More precisely, we study parabolic-elliptic simplification, or PES, varepsilonto 0^+ with fixed tau>0 up to the critical dimension N=4, and indirect-direct simplification, or IDS, (varepsilon,tau)to (0^+,0^+) up to the critical dimension N=2. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the L^p-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

Dynamic Mode Decomposition (DMD) is an equation-free method that aims at reconstructing the best linear fit from temporal datasets. In this paper, we show that DMD does not provide accurate approximation for datasets describing oscillatory dynamics, like spiral waves and relaxation oscillations, or spatio-temporal Turing instability. Inspired from the classical "divide and conquer" approach, we propose a piecewise version of DMD (pDMD) to overcome this problem. The main idea is to split the original dataset in N submatrices and then apply the exact (randomized) DMD method in each subset of the obtained partition. We describe the pDMD algorithm in detail and we introduce some error indicators to evaluate its performance when N is increased. Numerical experiments show that very accurate reconstructions are obtained by pDMD for datasets arising from time snapshots of some reaction-diffusion PDE systems, like the FitzHugh-Nagumo model, the lambda-omega system and the DIB morpho-chemical system for battery modeling.

We present a stability analysis of the standard nonautonomous systems type for a recently introduced generalized Lane-Emden equation which is shown to explain the presence of some of the structures observed in the atomic spatial distributions of magnetically-trapped ultracold atomic clouds. A Lyapunov function is defined which helps us to prove that stable spatial structures in the atomic clouds exist only for the adiabatic index γ=1+1/n with even n. In the case when n is odd we provide an instability result indicating the divergence of the density function for the atoms. Several numerical solutions, which according to our stability analysis are stable, are also presented.

The dynamics of cells, organisms, and fluids are often modeled as probability distributions evolving over time. Reconstructing and extrapolating this evolution from unpaired snapshots requires assumptions about the underlying process. Wasserstein gradient flows are a common choice, but they cannot describe conservative or periodic dynamics. Lagrangian mechanics in Wasserstein space covers both, but existing methods for learning it are simulation-based: they run a numerical solver at every training step, which makes training expensive. We propose Double-Stitch, a simulation-free method that learns these mechanics by penalizing the residual of the equation of motion along a learned population path. We derive this equation from a Clebsch variational principle that does not require gradient velocities, and show that the residual vanishes exactly when the equation holds. We test Double-Stitch on synthetic, single-cell and ocean vortex datasets and find that it matches or outperforms gradient-flow methods and simulation-based WLM on most tasks, while training 4-14 times faster than WLM. We provide a JAX implementation of Double-Stitch at https://github.com/BasisResearch/stitching.

We report a new system of artificial life called Lenia (from Latin lenis "smooth"), a two-dimensional cellular automaton with continuous space-time-state and generalized local rule. Computer simulations show that Lenia supports a great diversity of complex autonomous patterns or "lifeforms" bearing resemblance to real-world microscopic organisms. More than 400 species in 18 families have been identified, many discovered via interactive evolutionary computation. They differ from other cellular automata patterns in being geometric, metameric, fuzzy, resilient, adaptive, and rule-generic. We present basic observations of the system regarding the properties of space-time and basic settings. We provide a broad survey of the lifeforms, categorize them into a hierarchical taxonomy, and map their distribution in the parameter hyperspace. We describe their morphological structures and behavioral dynamics, propose possible mechanisms of their self-propulsion, self-organization and plasticity. Finally, we discuss how the study of Lenia would be related to biology, artificial life, and artificial intelligence.

In this paper we review the physical relevance of a Korteweg-de Vries (KdV) equation with higher-order dispersion terms which is used in the applied sciences and engineering. We also present exact traveling wave solutions to this generalized KdV equation using an elliptic function method which can be readily applied to any scalar evolution or wave equation with polynomial terms involving only odd derivatives. We show that the generalized KdV equation still supports hump-shaped solitary waves as well as cnoidal wave solutions provided that the coefficients satisfy specific algebraic constraints. Analytical solutions in closed form serve as benchmarks for numerical solvers or comparison with experimental data. They often correspond to homoclinic orbits in the phase space and serve as separatrices of stable and unstable regions. Some of the solutions presented in this paper correct, complement, and illustrate results previously reported in the literature, while others are novel.

We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters. These symmetries arise from the presence of one or more planes/axis of symmetry in the system's morphology, resulting in harmonious duplication and distribution of body parts. Significantly, we characterize how morphological symmetries extend to symmetries in the system's dynamics, optimal control policies, and in all proprioceptive and exteroceptive measurements related to the system's dynamics evolution. In the context of data-driven methods, symmetry represents an inductive bias that justifies the use of data augmentation or symmetric function approximators. To tackle this, we present a theoretical and practical framework for identifying the system's morphological symmetry group G and characterizing the symmetries in proprioceptive and exteroceptive data measurements. We then exploit these symmetries using data augmentation and G-equivariant neural networks. Our experiments on both synthetic and real-world applications provide empirical evidence of the advantageous outcomes resulting from the exploitation of these symmetries, including improved sample efficiency, enhanced generalization, and reduction of trainable parameters.

In an unbounded plane, straight lines are used extensively for mathematical analysis. They are tools of convenience. However, those with high slope values become unbounded at a faster rate than the independent variable. So, straight lines, in this work, are made to be bounded by introducing a parametric nonlinear term that is positive. The straight lines are transformed into bounded nonlinear curves that become unbounded at a much slower rate than the independent variable. This transforming equation can be expressed as a continued fraction of straight lines. The continued fraction is real-valued and converges to the solutions of the transforming equation. Following Euler's method, the continued fraction has been reduced into an infinite series. The usefulness of the bounding nature of continued fraction is demonstrated by solving the problem of image classification. Parameters estimated on the Fashion-MNIST dataset of greyscale images using continued fraction of regression lines have less variance, converge quickly and are more accurate than the linear counterpart. Moreover, this multi-dimensional parametric estimation problem can be expressed on xy- plane using the parameters of the continued fraction and patterns emerge on planar plots.

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equations

We investigate Josephson arrays consisting of a dice-lattice network of superconducting weak links surrounding rhombic plaquettes of proximitized semiconductor. Josephson coupling of the weak links and electron density in the plaquettes are independently controlled by separate electrostatic gates. Applied magnetic flux results in an intricate pattern of switching currents associated with frustration, f. For depleted plaquettes, the switching current is nearly periodic in f, expected for a phase-only description, while occupied plaquettes yield a decreasing envelope of switching currents with increasing f. A model of flux dependence based on ballistic small-area junctions and diffusive large-area plaquettes yields excellent agreement with experiment.

Prior studies on the emergence in large models have primarily focused on how the functional capabilities of large language models (LLMs) scale with model size. Our research, however, transcends this traditional paradigm, aiming to deepen our understanding of the emergence within LLMs by placing a special emphasis not just on the model size but more significantly on the complex behavior of neuron interactions during the training process. By introducing the concepts of "self-organization" and "multifractal analysis," we explore how neuron interactions dynamically evolve during training, leading to "emergence," mirroring the phenomenon in natural systems where simple micro-level interactions give rise to complex macro-level behaviors. To quantitatively analyze the continuously evolving interactions among neurons in large models during training, we propose the Neuron-based Multifractal Analysis (NeuroMFA). Utilizing NeuroMFA, we conduct a comprehensive examination of the emergent behavior in LLMs through the lens of both model size and training process, paving new avenues for research into the emergence in large models.

We prove stability for a class of heterogeneous catalysis models in the L_p-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.

We investigate soliton formation and interactions in the non-integrable Gardner-Ostrovsky equation (alias the rotation-modified Gardner equation) with anomalous (positive) dispersion. This equation is relevant to the description of wave dynamics in plasmas and other dispersive media. We show that a special class of solitons, characterized by zero total mass and non-monotonic asymptotics, can emerge from pulse-like initial perturbations. Depending on the initial conditions, these solitons may form regular amplitude-ordered trains, irregular nonstationary ensembles of interacting localized structures, or stationary propagating multi-soliton complexes. We further demonstrate that soliton interactions in the Gardner-Ostrovsky equation are inherently inelastic, leading to the emergence of a dominant "soliton-champion" in closed systems. For example, under periodic boundary conditions, only the largest-amplitude soliton ultimately survives, while all smaller solitons are gradually eliminated through successive interactions. Despite their inelastic interactions, Gardner-Ostrovsky solitons exhibit remarkable robustness.

In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle to efficiently capture regularities in the world, but the role of symmetry breaking is not well understood. Here, we develop a theoretical framework to study the "geometry of learning dynamics" in neural networks, and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. To build this understanding, we model the discrete learning dynamics of gradient descent using a continuous-time Lagrangian formulation, in which the learning rule corresponds to the kinetic energy and the loss function corresponds to the potential energy. Then, we identify "kinetic symmetry breaking" (KSB), the condition when the kinetic energy explicitly breaks the symmetry of the potential function. We generalize Noether's theorem known in physics to take into account KSB and derive the resulting motion of the Noether charge: "Noether's Learning Dynamics" (NLD). Finally, we apply NLD to neural networks with normalization layers and reveal how KSB introduces a mechanism of "implicit adaptive optimization", establishing an analogy between learning dynamics induced by normalization layers and RMSProp. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

We identify and formalize a fundamental gradient descent phenomenon resulting in a learning proclivity in over-parameterized neural networks. Gradient Starvation arises when cross-entropy loss is minimized by capturing only a subset of features relevant for the task, despite the presence of other predictive features that fail to be discovered. This work provides a theoretical explanation for the emergence of such feature imbalance in neural networks. Using tools from Dynamical Systems theory, we identify simple properties of learning dynamics during gradient descent that lead to this imbalance, and prove that such a situation can be expected given certain statistical structure in training data. Based on our proposed formalism, we develop guarantees for a novel regularization method aimed at decoupling feature learning dynamics, improving accuracy and robustness in cases hindered by gradient starvation. We illustrate our findings with simple and real-world out-of-distribution (OOD) generalization experiments.

Scaling limits, such as infinite-width limits, serve as promising theoretical tools to study large-scale models. However, it is widely believed that existing infinite-width theory does not faithfully explain the behavior of practical networks, especially those trained in standard parameterization (SP) meaning He initialization with a global learning rate. For instance, existing theory for SP predicts instability at large learning rates and vanishing feature learning at stable ones. In practice, however, optimal learning rates decay slower than theoretically predicted and networks exhibit both stable training and non-trivial feature learning, even at very large widths. Here, we show that this discrepancy is not fully explained by finite-width phenomena. Instead, we find a resolution through a finer-grained analysis of the regime previously considered unstable and therefore uninteresting. In particular, we show that, under cross-entropy (CE) loss, the unstable regime comprises two distinct sub-regimes: a catastrophically unstable regime and a more benign controlled divergence regime, where logits diverge but gradients and activations remain stable. Moreover, under large learning rates at the edge of the controlled divergence regime, there exists a well-defined infinite width limit where features continue to evolve in all the hidden layers. In experiments across optimizers, architectures, and data modalities, we validate that neural networks operate in this controlled divergence regime under CE loss but not under MSE loss. Our empirical evidence suggests that width-scaling considerations are surprisingly useful for predicting empirically maximal stable learning rate exponents which provide useful guidance on optimal learning rate exponents. Finally, our analysis clarifies the effectiveness and limitations of recently proposed layerwise learning rate scaling for standard initialization.