Papers

Echo State Networks (ESNs) are fundamentally important from the conceptual perspective of AI, because they provide simple and tractable frameworks for understanding temporal information processing, memory, and learning in dynamical systems. They are composed by an interconnected reservoir whose connectivity matrix remains constant over time and only the output weights are updated during the training period. In particular, ESNs employing online training allow the incremental adaptation of the output weights as data is received, making them particularly suitable for applications in which new input data becomes available over time (non-stationary environments). A key objective in the study of such learning structures is to determine which statistical properties of the reservoir enhance computational efficiency and maximize prediction performance. To this end, we investigate the effects of reservoir topology on learning performance, considering weight matrices generated from Erdos-Renyi random networks and Watts & Strogatz small-world networks. As test data we use the Lorenz and Rossler models in the chaotic regime and the FitzHugh-Nagumo model in the limit cycle regime. In the case of random Erdos-Renyi reservoirs, our simulations show that best performances are obtained for sparse reservoirs and especially when the link-density is near and immediately above the critical point of mean-field bond percolation. Watts & Strogatz reservoirs are shown to demonstrate best performance also for small values of link-density, compatible with the mean-field bond percolation when taking into account the specific nonlocal ring structure of the Watts & Strogatz reservoirs. These studies on the influence of the reservoir architecture and hyperparameters on enhancing the accuracy of ESN predictions provide an efficient framework for real-time forecasting in dynamically evolving systems.

Global synchronization has been largely investigated in static higher-order networks and little attention has been brought to directed, temporal higher-order networks. We hereby make a step in this direction by studying the emergence of global synchronization in periodically fast switching systems with directed higher-order interactions. By working in the framework of the Master Stability Function, we show that the linearized dynamics includes two time-dependent Jacobian matrices, both evolving on the timescale of the synchronous solution, together with a generalized Laplacian evolving on a fast switching timescale, resulting from the hypergraph time evolution. To address this two-timescale structure, we establish a fast-switching stability criterion for directed balanced higher-order structures. The latter condition allows to decouple the parallel and transverse perturbations to the synchronization manifold. By relying on the average theory for dynamical systems, we are able to prove exponential stability of global synchronization in the fast switching higher-order network. The proposed theory is then illustrated by using a higher-order Stuart–Landau model, which provides a natural benchmark for the theory. Directionality can move Laplacian eigenvalues into instability regions off the real axis, hence destabilizing synchronization; our results nevertheless reveal that temporal switching can counteract this destabilizing effect and restore global synchronization when the directed structures alternate sufficiently rapidly, even though the instantaneous directed higher-order structures are individually unstable.

Quantum aging transition has recently been reported, whereby the oscillatory dynamics of a network of globally coupled quantum self-sustained oscillators is degraded sharply when the fraction of inactive nodes exceeds a critical value. Here we explore the reverse process: the revival of rhythmic dynamics from an aged state in a quantum oscillator network by introducing a controlled incoherent single-boson pumping channel. Using a self-consistent quantum master equation in the Lindblad form for globally coupled active–inactive quantum Stuart–Landau oscillators, we show that increasing the pumping strength progressively suppresses the aging transition and eventually eliminates it, restoring rhythmicity even when all oscillators are inactive. Remarkably, complete compensation of the coupling-induced dissipative loss is not required; a substantially weaker pumping strength is sufficient to eliminate the aging transition. We show that the revival is governed by a redistribution of the Fock-state population of the inactive oscillators toward higher-occupation states. Our results establish controlled incoherent pumping as an effective mechanism for restoring dynamical robustness in quantum oscillator networks.

While the robustness of collective dynamics in static networks is well established, how agent mobility and network topology jointly shape dynamical robustness remains largely unexplored. In this letter, we investigate the robustness of mobile Stuart-Landau oscillators performing random walks on networks and interacting diffusively when co-located. In the rapid-mobility regime, we reduce the dynamics to a mean-field framework linking network degree statistics to an effective global coupling strength. We show analytically that increased degree heterogeneity in the network enhances the effective coupling strength, and strictly lowers the critical threshold of functional degradation for amplitude death. Numerical simulations confirm the predicted heterogeneity-dependent ordering of the transition thresholds, showing an earlier collective collapse in scale-free networks than in random or regular networks. Our results establish mobility and network heterogeneity as key determinants of robustness in mobile oscillator populations, with implications for resilient biological and engineered collectives.

Belief propagation in social networks depends on how trust and distrust jointly shape interpersonal influence. We develop a network model that treats trust and distrust as distinct but statistically coupled quantities. Their joint values determine the net trust and uncertainty of each relationship, and a neighbor can influence an agent only when both threshold conditions are satisfied. We show that the correlation between trust and distrust reshapes the transition between consensus and fragmentation by changing the availability of influential neighbors. Network connectivity also affects this transition, whereas degree heterogeneity has a more limited effect when networks have comparable mean degree. Moreover, the availability of influential neighbors alone does not determine the final consensus: correlations between relational properties and network degree can select agents with different degree compositions, altering the direction and magnitude of collective belief. Finally, we introduce a comfort index that quantifies the expected availability of influential neighbors and analytically links the joint trust-distrust structure to collective dynamics. Our results establish a mechanistic connection between relational structure, effective social influence, and emergent collective belief states.

Transport networks are key to living and engineered systems. How can transport networks adapt efficiently using only local rules, without a global objective or specific boundary conditions? Inspired by observations in biological transport networks, we adapt networks by subsequently pruning edges by relative resistance, thus removing edges whose resistance is higher than the resistance for routing flow through the rest of the network. We find that adaptation by such a local, boundary-condition-free measure of redundancy, set by topology alone, yields sparse, loop-rich, low-dissipation networks.

The role of adaptive coupling in controlling collective noise-induced dynamics is demonstrated on an example of stochastic resonance in an ensemble of coupled overdamped bistable oscillators. Specifically, it is found that tuning the adaptive coupling parameters allows one to either enhance or suppress stochastic resonance and to shift the optimal noise intensity that yields the most regular stochastic oscillations. All the revealed effects are studied in numerical simulations for two kinds of adaptive coupling models. The first one is a simplified, phenomenological model designed such that the coupling becomes stronger for oscillators with larger differences between their dynamical variables (two configurations for local interaction and one option for global coupling are under consideration). The second one is a local memristive coupling model that provides a physically motivated realization of adaptive interactions with similar properties. In all the considered cases, adaptive coupling is shown to be a flexible stochastic resonance control tool, both in the absence and in the presence of the static coupling component.

Theories of how the brain processes information and returns intelligent outputs are numerous and often difficult to conclusively test. Here, we consider an approach grounded in informatics and algorithmic thermodynamics for how neural systems respond to different patterns of information in different ways, with a focus on the relationship between the information entropy of a signal and a new proposed quantity we term information enthalpy, which represents the internal resource of structured information within a system available to be used for predictions and other forms of work. To evaluate the capacity of the external signals to increase information enthalpy in the system, we introduce the information enthalpy potential (IEP) as a defined metric. We propose and implement a method for quantifying the amount of potential information enthalpy - the IEP - in a given signal across a range of example signals. We offer a conjecture of how information enthalpy may be treated by neural systems within a biological framework before showing how it may integrate and offer falsifiability to existing approaches such as the Free Energy Principle. This framework is also positioned in light of the role of neural criticality. Finally, we postulate a testable and falsifiable framework where the distinct mechanisms within each driver interact through a described n-body-inspired model to govern the "motion" of a neural system through a high dimensional state-space. Through these processes, it is proposed that these features form the most fundamental basis of the neural drivers that give rise to the complexity of adaptive behaviors that are commonly called: intelligence.

Live-streaming platforms generate high-frequency records of collective human activity in which many loosely coordinated users react in real time to shared external stimuli and to one another. We investigate whether the temporal organization of Twitch chat messages exhibits multifractal signatures characteristic of complex dynamical systems. Selected high-activity chats were converted into time series by measuring the length of each consecutive message in words, with platform emotes treated as individual tokens. The resulting signals display heterogeneous fluctuations, bursty organization, and heavy-tailed distributions that, in several cases, are better captured by q-Weibull forms than by conventional light-tailed models. Autocorrelation analysis reveals persistent temporal dependence over broad ranges of message lags, indicating that the variability is not produced by independent message generation alone. MFDFA shows that some chats possess well-developed multifractal spectra and generalized Hurst exponents strongly dependent on moment order, whereas others exhibit weak scaling, approximately monofractal behavior, or no well-defined scaling regime. Surrogate tests indicate that temporal correlations are essential for the observed multifractality, while heavy tails enhance its apparent strength. Under particular dynamical conditions, Twitch chat may therefore form an emergent collective signal whose fragmented, reactive, associative, and multiscale organization is structurally reminiscent of stream-of-consciousness narrative. This expression is used strictly as a structural and dynamical analogy, not as a claim of a unified collective mind or of semantic, cognitive, or phenomenological equivalence with literary stream-of-consciousness writing. The results position live-stream chat dynamics within the broader physics of complex, intermittent, human-generated signals.

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.

This study considers rapidly modifying the wake shedding frequency of the flow around an airfoil using sparse sensor information, subject to constraints on the lift coefficient fluctuations. This is achieved by combining phase-amplitude reduction with nonlinear machine-learning-based sparse sensor reconstruction. We derive time-varying phase and amplitude sensitivity fields that identify the optimal spatial locations and timing for actuation from merely three sensors. Through the sensitivity fields, we analytically obtain the optimal waveform for fast synchronization of wake shedding frequency while minimizing amplitude deviation of aerodynamic responses. The proposed approach is evaluated using flows over various NACA airfoils at several post-stall angles of attack, all of which exhibit unsteady periodic vortex shedding. With the identified optimal forcing, the wake frequency is altered much faster than with a standard sinusoidal actuation. Furthermore, the amplitude-penalized forcing achieves $20\%$ suppression of the lift coefficient fluctuation compared to the optimal forcing without amplitude penalty. The current amplitude-penalized technique may offer an efficient path for fast flow modification without causing detrimental fluctuations in periodic aerodynamic and aeroelastic systems with fluid-structure interactions.

Body-body interactions have long been investigated to leverage the performance of various fluid-based machines. This study analyzes the flow interaction between a cylinder and a NACA0012 airfoil using direct numerical simulation and machine learning. We consider a two-dimensional incompressible flow at Reynolds number 100, where the airfoil is positioned in the wake of a cylinder. By sweeping the parameter space, composed of the distance between the cylinder and the airfoil $\Delta x$, the relative height $\Delta y$, and the angle of attack of the airfoil $\alpha$, four characteristic vortex-shedding regimes are observed. This study particularly focuses on suppressed vortex shedding at a certain arrangement as oscillation death in coupled oscillators, with the theoretical prediction of flow stability on a machine-learned low-order coordinate. Independent flows around each body are regarded as a system that exhibits isolated periodic vortex shedding at infinite distance, and interactions appear at smaller distances. We employ a synchronization-theoretic autoencoder to derive models for two coupled oscillators from an aerodynamic coefficient. The proposed approach, which enables the assessment of system stability and the prediction of vortical interaction behavior, may pave the way for the analysis of unsteady body-body interactions from the data-driven and synchronization perspectives.

Mean-field games provide a continuum framework for modeling the dynamics of large, interacting populations of non-cooperative agents. This paper studies symmetry-breaking in a finite-horizon, two-choice min-LQG mean-field game in which identical agents with linear stochastic dynamics choose one of two equally desirable terminal destinations, while trading off control effort against social pressure to conform. The model has an odd symmetry between the two destinations and therefore always admits a symmetric, dynamic Nash equilibrium in which the population splits evenly between them, producing a deadlock collective state at final time. Numerical studies have suggested that, as the penalty for social nonconformity increases, this symmetric equilibrium loses stability as a fixed point of an associated scalar, self-consistency map, and asymmetric consensus Nash equilibria emerge, where most agents select the same destination. By analyzing the linearized forward-backward PDE system through the scalar map representation, we provide a proof of this loss of stability of the symmetric equilibrium. Together with the odd symmetry of the map, this implies the existence of symmetry-broken mean-field game equilibria corresponding to consensus on either destination.

Across biological and social complex systems, models of epidemics or contagions incorporate complexity through network correlations, adaptive behavior, higher-order interactions, or co-circulating strains. Tracking these mechanisms lead to high-dimensional mathematical systems that resist analytical treatment. We show that a broad class of models admits a systematic reduction via fast-variable elimination. The central insight is that complexity can be captured by fast variables equilibrating much faster than the epidemic, leading to “emergent complex contagion” where the force of infection depends on prevalence. The epidemic transition follows a universal equation and is continuous or discontinuous whether the force of infection decays or intensifies with prevalence. Our results reveal diverse epidemic models share a common skeleton such that mechanisms that share a characteristic form can be macroscopically indistinguishable.

Collective behavior in animal groups is routinely described in cognitive language: fish schools sense gradients, swarms compute, colonies decide. These formulations often leave open what the collective is inferring, what computational object is represented, or how that representation arises from microscopic interactions. Here we develop that description quantitatively in a simulation based on the canonical model of collective sensing in fish schools (Berdahl et al., 2013), in which collective gradient sensing emerges from individuals that do not estimate gradients themselves: agents only modulate their speed in response to local light intensity and respond to simple social forces from neighbors. We show that these local rules admit a mesoscopic Bayesian reinterpretation at the level of the school, and hypothesize that the collective dynamics realize approximate inference on a generative model of how darkness gradients drive inter-individual speed differences. The resulting school-level inference process yields a posterior distribution over the local darkness gradient, whose statistics vary with environmental structure and relate systematically to collective motion and sensing performance. The school can therefore be described, in a Bayesian-mechanical sense, as a distributed inferential system without requiring individual agents to perform probabilistic belief updates themselves. This yields a quantitative, falsifiable account of collective sensing as emergent inference, and a concrete bridge from collective behavior to Bayesian mechanics.

Griffiths phases provide a mechanism for sustaining critical dynamics over extended parameter regions without fine-tuning to an isolated transition point. This behavior has been mainly associated with structural heterogeneity in complex networks. Here we show that quenched heterogeneity in local parameters can generate a chaotic Griffiths phase in a network of map-based neurons. We study globally coupled Chialvo maps with quenched disorder in a local neuronal parameter, thereby isolating intrinsic parameter disorder from topological disorder. As the coupling strength is varied, a chaotic Griffiths phase emerges between desynchronized and coherent chaotic dynamics, characterized by intermittent formation and breakup of neuronal clusters and large fluctuations in the fraction of clustered neurons. Within this regime, cluster sizes exhibit power-law distributions with coupling-dependent exponents, providing evidence of critical behavior over a finite parameter interval. The chaotic Griffiths phase is absent in the homogeneous limit and broadens progressively as the degree of parameter heterogeneity increases. We further introduce topological disorder through small-world connectivity and find that its combined action with local parameter heterogeneity substantially extends the chaotic Griffiths phase. In both globally coupled and small-world networks, the number of positive Lyapunov exponents displays anomalous scaling with system size. These results demonstrate that intrinsic dynamical heterogeneity constitutes an independent mechanism for generating chaotic Griffiths phases and that its interplay with structural disorder enhances extended criticality in neuronal networks.

We investigate how root allometry and below-ground root competition due to root overlap affect the conditions for pattern formation in non-local scalar vegetation models. We start from a spatially explicit two-component vegetation-water system in which plants acquire soil water through their laterally extended root systems, with root allometry represented by an interaction kernel whose spatial extent increases with the above-ground biomass density. In the limit of fast water dynamics, we eliminate the water variable, resulting in a non-local scalar model for the vegetation dynamics. This reduction reveals that accounting for root overlap naturally leads to a double integral structure in the scalar model, in contrast to the single convolution term commonly used to model non-local competition for resources. For the resulting scalar model, we identify two distinct mechanisms that can drive Turing instabilities. The first instability mechanism requires root growth to outpace mortality, while the second mechanism relies on the development of competitive pressure zones, in which vegetation growth is impeded by the cumulative competitive pressure exerted by neighbouring areas with higher biomass density. Root allometry promotes the occurrence of both instability mechanisms, while root overlap inhibits the emergence of competitive pressure zones. Hence, our results show that both root allometry and root overlap are important features to consider when studying vegetation pattern formation.

We study a high-dimensional Kuramoto model with attractive pairwise coupling and a repulsive higher-order effect. Although the pairwise interaction favors synchronization, the higher-order interaction prevents complete synchronization and selects an intermediate level of coherence corresponding to the balanced equilibria. For the unit sphere of dimension at least two, we provide an explicit basin of attraction for the balanced equilibria. We also classify nonzero-mean equilibria and show that all non-balanced equilibria are linearly unstable. This indicates that balanced states are the only natural stable candidates in the higher dimensions. In contrast, on the circle, the same model reduces to a higher-order Kuramoto-type model which exhibits a qualitatively different selection mechanism that depends fundamentally on the dimension. In this case, population imbalance at finite $N$ prevents exact stationary balance and instead generates a common angular drift producing phase-locked states. We construct basins of attraction for two-cluster locked states, in which the phases split into two groups, and identify the corresponding finite-$N$ selected values of the order parameters. This phenomenon is called continuous-versus-quantized order-parameter selection. We further demonstrate, through a root-selection mechanism, why such two-cluster locked states are typically observed in most simulations. Finally, we show that two-cluster locked states with a large population imbalance are linearly unstable, which explains why only certain locked states are dynamically robust.

Extreme events are widely studied within simple random walk frameworks, where their probability is determined by the network structure and stationary walker distribution. Here, we propose a recovery random walk (RRW) model in which extreme events temporally `freeze' the nodes where they occur for a fixed duration $\Delta$ ($\Delta=0$ recovers the original model), trapping walkers and reducing the effective mobile population, thereby making the model more practical. We derive a first-principles description of this feedback and a delayed differential equation for the frozen-node fraction. This finite freezing produces an initial overshoot, followed by damped oscillatory relaxation to a steady state for the fraction of the frozen nodes. We find that freezing suppresses extreme event probability while preserving its degree dependence dynamics, and suppresses the EE frequency. Further, the analytical prediction for the fraction of the frozen nodes agrees closely with simulations. This model closely reflects real-world scenarios, yielding more practical EE statistics.

Global Dust Storms (GDSs) dominate the Martian climate, yet the mechanisms of dust lifting, transport, and deposition remain poorly constrained because in situ observations are limited: landers are stationary, and their Radioisotope Thermoelectric Generators (RTGs) contaminate local measurements. Our solution is a swarm of lightweight, wind-propelled spherical rovers – Dust Rovers (DRs) – that follow natural wind patterns to collect high-spatial-resolution in situ dust data during storms. Here, we quantify the dispersion and communication connectivity of this robotic architecture. Using 625 wind states calculated using the Mars Climate Database (MCD) for the MY34 storm at the Curiosity site, we build a bivariate-normal model of the near-surface wind velocity and characterise its diurnal structure. Rover trajectories and swarm dispersion are propagated with three integrators: the Euler and fourth-order Runge-Kutta schemes for the deterministic dynamics, and a stochastic Ornstein-Uhlenbeck formulation that simulates turbulent gust forcing. Dispersion is controlled by the unmeasured gust correlation time: with a static rolling threshold enforced, the white-gust case (tau_g = Delta t = 1 s) provides a two-rover separation below 1 m over an hour (0.44 km over a sol), whereas correlation times of 10-60 s yield 6-25 m over an hour (1.4-3.3 km over a sol). We demonstrate that a 20-rover swarm needs only a 1.6-4.1 km per-rover range to remain connected over a sol, and stays connected across the plausible envelope – for gust amplitudes up to twice the nominal value with tau_g <= 60 s, and at the nominal amplitude up to tau_g ~ 150 s.

Emergence, features of a whole that no part shows alone, is central to complex systems but rarely measured so as to locate its source. Viewing a system structurally, as a network of interacting parts, we evaluate emergence as what observing the coupled whole reveals beyond, or erases from, the combined observations of the parts. For observations that build what they see route by route, we show that every emergent feature is produced by its own route across the interface between parts, a route that traces it to the components and interactions responsible. We show the number of all such boundary-crossing paths bounds the total emergence discrepancy, and equals the emergence of measures that count routes. The bound is computed locally around the interface, validated through network examples, and in a nervous system the routes show that interneurons relay most of the emergence. Our work associates emergence with path structure in the network system that can be anticipated, attributed, and engineered.

We study the delayed one-dimensional swarmalator model when all units share a common intrinsic frequency {\omega}. Unlike the zero-frequency case, {\omega} cannot be removed by a rotating-frame transformation because delayed interactions retain the phase accumulated over the lag. The result is a two-order-parameter analogue of the delayed Kuramoto model: the asynchronous state acquires incoherence lobes, the synchronized state becomes a rotating branch stable even when both the spatial and phase couplings are repulsive, and the phase wave develops narrow resonant stability bands. These delay-selected windows overlap to produce broad branch coexistence, including a narrow window near K=-J where all three canonical states-asynchronous, phase wave, and synchronized-are simultaneously stable. In regions where no canonical branch is stable, the order parameters oscillate persistently.

Time series prediction is a common application of reservoir computing. When the training and testing time series data contains multiple dynamical regimes, because an underlying parameter is changing, or the data in fact consists of multiple distinct systems, simple application of the reservoir computing principle produces high prediction errors. Here, we propose a HyperReservoir as an extended model of reservoir computing especially designed for such cases. The HyperReservoir combines a main reservoir with a smaller context reservoir, where the latter modulates the output weights of the former. This structure resembles the hypernetworks from deep neural network literature. However, in contrast, HyperReservoirs retain the simple training via linear regression of standard reservoir computing. We compare the proposed architecture with a conventional ESN, in which context acts at the input, and a full-matrix Conceptor, in which context modulates the reservoir state space. We evaluate all three models on time-series prediction tasks based on Lorenz and Rössler systems, including for varying bifurcation parameters and time sampling scales. We find that the HyperReservoir achieves the lowest mean test error in all three tasks, and particularly outperforms conceptors on data that is sampled from the same attractor but at different time scales.

Topological signals are dynamical variables supported on higher-order structures such as simplicial or cell complexes. In this work, we investigate the impact of time-delayed interactions on the emergence of global synchronization of topological oscillators, with application to the Stuart-Landau system. We first examine the case where signals are supported on simplexes, or cells, of a given dimension and coupled through the Hodge-Laplace matrix. We derive the Master Stability Function and we prove that for suitable discrete delay values, the stability of the synchronous solution becomes analytically tractable and can be solved by using the Lambert W-function. This analysis yields explicit spectral conditions for Global Topological Synchronization to emerge and shows that admissible delays may either promote or suppress synchronization. We then study delayed interactions between signals supported on simplexes, or cells, of different dimensions coupled by the Dirac operator. Under analogous admissible time delays the variational problem, still related to the Master Stability Function, can again be simplified, however in this case we prove that Global Topological Dirac Synchronization cannot emerge. Numerical simulations on simplicial and cell complexes corroborate and complement these findings.

We examine the spectral dissipation-coherence trade-off inequality conjectured by Oberreiter et al. [Phys. Rev. E 106, 014106 (2022)], stating that the coherent number, defined from the second largest eigenvalue, provides a lower bound on the stationary entropy production per unit oscillation. We disprove this conjecture by explicitly constructing a counterexample, which accompanies finite coherent number with vanishing stationary entropy production. This model also excludes a wide class of thermodynamic bound with the real and imaginary parts of the second largest eigenvalue of the transition rate matrix. Our result implies that the second largest eigenvalue does not necessarily characterize the degree of coherent oscillation.

The one-dimensional swarmalator model is usually understood through three macroscopic states: synchrony, the phase wave, and asynchrony. We show that the identical model also contains a much larger set of hidden stable equilibria - families of q-twisted and m-clustered states - which random initial conditions almost never reach, because many are embedded in a degenerate neutral manifold of vanishing global basin. They are nonetheless locally robust: under Gaussian perturbations of per-coordinate amplitude {\epsilon} their survival radii shrink only algebraically, as 1/q and 1/m, in sharp contrast to the Gaussian basins and winding ceiling of twisted states on a nearest-neighbour Kuramoto ring, both removed by mean-field coupling. A block-circulant Jacobian fixes the stability in closed form, and far-kick experiments indicate the basins are compact cores rather than tentacled sets. Several patterns reported in extended swarmalator models thus already exist, stable or neutrally stable, in the minimal one.

We develop an analytical framework for studying a family of interacting particle systems that we refer to as the multi-particle Oja flow. The common interaction structure underlying this family arises in a variety of applications, including higher-order synchronization models, opinion dynamics, and self-attention dynamics derived from Transformer architectures, motivating a unified mathematical framework for their analysis. We first establish well-posedness for both the finite-particle and the mean-field systems. We also prove convergence of the finite-particle dynamics to the mean-field equation as the number of particles tends to infinity. We then investigate low-dimensional structures underlying the dynamics. In particular, we show that the mean-field solution admits a representation through a time-dependent normalized linear transformation acting on the initial distribution, with the transformation governed by a matrix ODE. Building on this representation, we identify a broad class of parametric families that are invariant under the mean-field dynamics. When the initial distribution belongs to one of these families, the evolution remains within the same family for all time and can therefore be characterized through the dynamics of the corresponding family parameters. As a concrete example, we develop this reduction within the angular central Gaussian family and illustrate how the resulting parameter dynamics provide reduced descriptions of the mean-field evolution for several representative systems.

Criticality in the Ising model is conventionally generated by Hamiltonian dynamics and controlled by temperature. Here we show that critical like behavior can emerge in an Ising system whose dynamics is completely independent of the Ising Hamiltonian and contains no temperature like parameter. We introduce an adaptive cluster dynamics in which spin connectivity is controlled by a local quiet time variable, the time elapsed since a spin was last updated. The Ising Hamiltonian enters only as an observable characterizing the resulting configurations. As the parameter $\alpha$, controlling the connectivity strength, is varied, the system undergoes spontaneous symmetry breaking accompanied by strong collective fluctuations. The cluster size distribution develops an extended power law regime, terminated by a small finite size hump at the largest scales, while the Ising energy exhibits a singular response to the variation of $\alpha$. These results show that memory dependent connectivity alone can generate collective critical behavior, revealing a route to nonequilibrium criticality without energy based dynamics.

Epidemic spread and collective opinion formation are tightly coupled processes: individual beliefs about the severity of an outbreak and the value of protective measures shape behavior, while the evolving epidemic itself reshapes public sentiment through risk perception. We propose an agent-based framework that couples opinion dynamics with a compartmental epidemic model, in which each agent carries both a health status and a continuous opinion variable describing adherence to protective measures. Opinion evolution is driven by binary interactions among agents and by an exogenous background field that reflects the current state of the epidemic, while the disease transmission rate depends explicitly on the opinions of interacting agents. Using methods of statistical mechanics, we characterize the resulting macroscopic system for the compartmental mass fractions and mean opinions in both far-from-equilibrium and close-to-equilibrium regimes. The calibration shows that incorporating behavioral opinion dynamics substantially improves the interpretation of epidemic evolution. In particular, the inferred risk perception function evolves consistently with the observed epidemic burden, indicating that behavioral adaptation can explain part of the temporal variation that would otherwise be absorbed into time-dependent transmission parameters. Our results illustrate how integrating opinion heterogeneity in kinetic epidemic models offers quantitative insights into the feedback between behavior and disease spread.

We show that negative feedback alone generates the diversity of self-organized shapes usually attributed to scale-dependent activation-inhibition. For a broad class of kernels, pairwise competition models only generate hexagonal spot arrays. Higher-order terms eliminate this restriction and promote stripes and gaps. Combining individual-based simulations and nonlinear analysis, we derive the pattern-selection thresholds and construct the full state diagram, including an unusual spots-stripes-spots sequence. Spot patterns are thus not a reliable indicator of proximity to a tipping point.