We investigate the most probable transition path from a perennially ice-covered state to an ice-free Arctic state, using a piecewise-smooth periodically forced energy balance model for the yearly evolution of energy in the Arctic introduced by Eisenman and Wettlaufer. Although a summertime ice-free state is widely expected to occur if current climate change trends persist, the time frame and pathway by which the Arctic may transition to an ice-free state remains in debate. Using the Freidlin-Wentzell theory of large deviations with an appropriate rate functional for piecewise-smooth systems, we characterize noise-induced transitions in the energy balance model framed as a stochastic differential equation. We derive the most probable transition path using gradient flow applied to a mollified system and validate our qualitative results using Monte Carlo simulations. These simulations show that transitions typically begin during the first half of the year with an average duration of approximately 19.5 years. We further derive an effective quasi-potential to estimate the expected escape time from the metastable states to the switching manifold. Our results provide insight into Arctic sea ice loss mechanisms, as well as a case study for how transition paths may be characterized in piecewise-smooth systems.
In this paper, we study the connections between two saddles in a rotated vector field, where one saddle lies in R2 and the other is located at infinity. We provide sufficient conditions for the existence of a heteroclinic connection and also of infinitely many heteroclinic connections between two saddles, one in the finite plane and another at infinity. These results are applied to a degenerate Bogdanov-Takens system with symmetry and to a cubic Liénard system, yielding complete global phase portraits on the Poincaré disk.
Discrete-time spiking neural networks provide an important framework for modeling discrete memristive synapses and for investigating their spike transmission and network dynamical behaviors. However, existing studies mostly focus on single-state discrete memristive synapses, and systematic research on second-order discrete memristive synapses in discrete-time spiking neural networks is still relatively scarce. Based on this, this paper constructs a second-order discrete memristive synapse model regulated by two internal-state variables in a coordinated manner. First, the basic dynamic characteristics of the proposed synapse model are analyzed, and its state evolution features and numerical stability performance under random spike input conditions are investigated. Subsequently, the model is embedded into a typical discrete neuron model for unified verification, and the results show that it can achieve stable pulse transmission. Furthermore, the influence of memristive synapse parameters and coupling strength on synchronization behavior is examined at the network level. The results indicate that the system can evolve from incomplete synchronization to high synchronization and exhibit chimera-like states in a ring-coupled network. Additionally, this paper conducts pulse coding, limited pattern discrimination, and spike-timing-dependent plasticity analysis based on the proposed synapse model. The results show that the model can maintain a certain information representation and pattern-discrimination ability under noise interference, providing a feasible solution for the stable modeling of memristive synapses and their information-processing applications in discrete spiking neural networks.
The computational efficiency of contemporary multi-scale digital twins used to investigate cardiac arrhythmia is often hindered by the equation system complexity, while the accuracy of predictions depends on the calibration of model parameters and their intricate non-linear relation. Here we employed a robust pipeline for fast simulations of scar-related ventricular tachycardia (VT) inducibility using an efficient Lattice-Boltzmann Method with GPU-based accelerated code. We tested the pipeline on 3D digital twins built from MR images acquired in n = 8 swine with chronic infarction, by performing a subject-specific personalization of each model per tissue type (i.e., scar, border zone, and healthy) and tuning the key parameters (e.g., action potential duration, excitability, and wave speed) from recorded endocardial bipolar voltage maps and intracardiac electrograms. We first validated the VT simulation outcome by precisely replicating the stimulation protocol and pacing site used in the animal studies. Our results demonstrated very good agreement between the experiment and simulated VT outcome for personalized model parameters using subject-specific values, compared to a poor outcome when parameters were tuned from average values from all cases. Second, we performed a comprehensive in silico study where the stimulation was delivered from 10 000 virtual endocardial locations and found a strong dependence of VT (non)/-inducibility per case on the stimulation site. Simulating 10 s of sustained VT induced on a 3D model comprising 1.2 million cubic voxels (0.7 mm edge size) took 30 min on a laptop using GPU, underlying the computational tractability of our pipeline and its potential clinical translation to predict infarct-related VT risk.
Cooperation is difficult to sustain in public-goods dilemmas because contributing can reduce an individual’s immediate payoff, while benefits to the surrounding group may emerge only after several rounds. We introduce a dual-value reinforcement-learning model for a spatial public goods game that evaluates these two consequences separately. One value system learns from the individual payoff obtained after each action, whereas the other learns from local welfare accumulated over multiple rounds. The two evaluations are combined only when an action is selected. Within the tested settings, numerical simulations show that cooperation is best supported when neither evaluation fully dominates. A moderate welfare-feedback horizon produces higher cooperation, fewer strategy changes, and stronger agreement between the two value systems than one-step or excessively long feedback. The results further show that increasing the influence or duration of social evaluation does not improve cooperation indefinitely. Stable cooperation instead depends on coordinating immediate individual incentives with delayed neighborhood-level consequences, rather than replacing payoff-oriented learning with social valuation.
This paper investigates the spatiotemporal dynamics of a hyperbolic reaction-diffusion predator-prey system with inertial effects. We first derive the critical conditions for codimension-one bifurcations (Hopf and Turing bifurcations) and codimension-two bifurcations (Turing-Turing and Turing-Hopf bifurcations). Theoretical and numerical results show that asymmetric inertial effects fundamentally alter the instability mechanism of the system: under equal diffusion rates, predator inertia alone can induce wave instability and self-organized spatiotemporal oscillations, whereas prey inertia mainly plays a stabilizing role. In addition, the multiple-scale method is successfully extended to the codimension-two bifurcation analysis of hyperbolic reaction-diffusion systems, overcoming the dimensional reduction difficulties encountered by the classical center manifold theory in dealing with higher-order time operators. The resulting normal forms accurately characterize the competition, selection, and transition of spatiotemporal patterns near the Turing-Hopf critical point. This study reveals the profound influence of inertial effects on the complex dynamics of nonlinear diffusion systems.
Randomly connected neural networks undergo a transition from a stable fixed point to chaos as the coupling strength increases. In the thermodynamic limit, this transition has been shown theoretically to occur abruptly at a critical point. In finite-size systems, however, a variety of bifurcation cascades appear between the stable fixed point and chaos. In this study, we systematically characterize routes to chaos in finite-size random neural networks. By analyzing individual realizations, we identify multiple scenarios, including the Ruelle-Takens-Newhouse route, torus doubling, and fractalization, as well as chaotic dynamics with persistent toroidal geometry. We also study these behaviors at the ensemble level by quantifying the fraction of chaotic trajectories as a function of coupling strength and system size. The resulting finite-size crossover sharpens with increasing system size and exhibits empirical scaling trends, providing a statistical characterization of the broad onset of chaos in finite random networks.
Factor models provide a representation of the joint behavior of large cross sections of financial assets through a reduced set of underlying drivers. In this work, we propose a network-based dynamical framework in which statistical factors emerge endogenously from interactions among assets, rather than being imposed exogenously or extracted purely statistically. We model asset returns as a system of coupled nonlinear maps influenced by a network of interactions encoded in a coupling matrix. This matrix is constructed via an orthogonal transformation of a Laplacian operator with prescribed nullity, allowing us to control the system’s effective dimensionality. Under appropriate coupling conditions, the dynamics reduce from a high-dimensional phase space to a lower-dimensional invariant manifold, where synchronization modes of co-movement arise. We show that the number of emergent statistical factors is directly related to the nullity of the coupling matrix, consistent with a stability analysis performed here. Simulations demonstrate that the model reproduces key stylized features of financial returns while generating factors with balanced loadings across assets. Finally, we show that this model reproduces features that are observed in an empirical portfolio. These findings suggest that statistical factors can be interpreted as emergent synchronization modes of an interacting nonlinear system, providing a complementary perspective to the existing economic and statistical factor models.
Online discussions often exhibit multidimensional polarization, in which stances on distinct topics become aligned along broader ideological dimensions. Analysis of Reddit data indicates that this cross-topic stance coupling appears within co-participating discussion groups. Motivated by this observation, we introduce a multidimensional opinion dynamics model on a two-layer structure with pairwise edges and hyperedges, together with a group conformity exponent that controls nonlinear group influence. Mean-field and bifurcation analyses reveal two dynamical mechanisms. Linear group conformity lowers the critical threshold at which neutral consensus loses stability and increases cross-topic coupling. Nonlinear group conformity leaves this threshold unchanged but introduces a subcritical bifurcation with a bistable regime. Empirical analysis of data from 27 subreddits spanning four ideological dimensions provides supporting evidence for these mechanisms and shows that the proposed model more closely reproduces observed polarization and coupling patterns than several benchmarks. These results highlight the role of group structures in shaping collective opinion dynamics on online platforms and may offer potential insights for platform governance.
We analyze excitation and relaxation dynamics in a two-dimensional heterogeneous excitable medium described by a smoothly regularized three-variable Fenton-Karma model. To relate spatial heterogeneity to a global observable, we examine the dipole response together with the evolution of two activation-threshold manifolds associated with distinct regimes of the excitation-relaxation dynamics of the model. We show that increasing diffusion contrast does not produce a monotonic change in the global excitation-relaxation organization. For weak contrast, excitation remains globally connected despite deformation of the excitation pattern. For intermediate contrast, the system fails to establish a globally connected excited state outside the heterogeneous region, leading to the disappearance of the characteristic negative T-wave-like minimum in the dipole signal. However, a further increase in diffusion contrast restores global connectivity despite the persistence of a trapped low-potential region within the heterogeneity. These results demonstrate that localized heterogeneity can induce a nonmonotonic reorganization of excitation-relaxation dynamics by modifying the ability of the system to establish globally connected excitation. Simultaneously, the global dipole signal directly encodes transitions between regimes with and without a globally connected excited state.
The reporting phenomenon is ubiquitous in human society. However, how reporting-triggered institutional incentives influence public cooperation remains theoretically unresolved. Here, we construct a public goods game where individuals adopt one of three strategies: report, cooperate, and defect. In this game, cooperators and reporters contribute a cost to the common pool, whereas defectors make no contribution. Furthermore, each reporter pays a monitoring cost to detect defection and subsequently report it to the institution. Upon receiving a report, the institution will punish defectors and reward reporters with a response probability, irrespective of the number of reporters. We introduce the proposed game into a structured population and analytically derive the average fractions of cooperators, defectors, and reporters in the weak-selection limit by using the coalescing random walk method. We demonstrate that increasing the institutional response probability, reward intensity, or punishment intensity elevates the average fraction of reporters and reduces that of defectors, while keeping the average fraction of cooperators invariant. We further determine the conditions under which reporters are favored by natural selection, reporters are favored over cooperators, and contributors, including cooperators and reporters, prevail, respectively. We find that only when institutional incentives adequately offset the costs of contribution and monitoring can reporters gain an evolutionary advantage and contributors become dominant in the population. Notably, our main results remain robust under a group-dependent implementation rule, where the probability of implementing institutional incentives depends on the number of reporters in the group. Finally, these theoretical predictions are validated by numerical simulations.
We investigate memory-induced collective dynamics in a ring of electrically coupled Hindmarsh-Rose neurons operating in a chaotic bursting regime. The effects of long-range interactions, diffusion delay, reaction delay, and electromagnetic induction on spatiotemporal organization and Hamiltonian energy density are systematically analyzed. The dynamics is quantified using five complementary statistical indicators: the synchronization factor, the strength of incoherence, the discontinuity measure, the skewness measure, and the time-averaged Hamiltonian energy density. Depending on the memory parameters, the network exhibits asynchronous, chimera, multichimera, and synchronized states. We show that long-range interactions increase the Hamiltonian energy density and enhance the level of collective activity, although they predominantly generate asynchronous states when acting alone. Diffusion delay generates a rich variety of collective states, ranging from asynchronous and chimera patterns to complete synchronization depending on the interaction range, while reaction delay predominantly supports asynchronous, chimera, and multichimera dynamics. In contrast, electromagnetic induction promotes global synchronization from a critical memristive coupling threshold. Furthermore, the Hamiltonian energy density increases monotonically with the interaction range, reaction delay, and memristive feedback strength, whereas it exhibits a non-monotonic dependence on diffusion delay. These findings demonstrate that multiple memory mechanisms interact synergistically and competitively to shape neuronal coordination, energy localization, and pattern formation, providing new insights into the design of robust neuromorphic systems and brain-inspired algorithms.
Traditional evolutionary game theory often represents population structure with dyadic networks, although many applications require payoffs defined over groups. Hypergraphs and simplicial complexes retain group membership explicitly, but their effects must be distinguished from nonlinearities already present in multiplayer payoff functions. This structured scoping review separates these two sources of variation. A multiplayer nonlinear-payoff effect arises from thresholds, synergy, saturation, or heterogeneous group benefits on a fixed interaction structure. Such effects can occur in well-mixed or spatial groups without higher-order topology. A higher-order topological effect arises from explicit group membership, hyperedge overlap, inter-order organization, or simplicial closure. Its assessment requires the payoff, update rule, group-size distribution, and selection regime to be fixed or closely matched. A documented search through July 30, 2026 identified 85 core works: 80 primary studies and 5 conceptual or review articles. Across this evidence, nonlinear payoffs generate frequency dependence, internal equilibria, invasion thresholds, and, in some models, multistability. Higher-order topology instead changes group sharing, spillover between groups, and the routes to invasion or fixation. Discontinuous transitions and hysteresis are model outcomes whose origins depend on the complete payoff, topology, update, and feedback specification. Related graph-based research on fairness, trust, AI safety races, open-data stewardship, and cheap talk defines a broader agenda. Social welfare must also be evaluated separately from cooperation frequency. Direct empirical tests and factorial comparisons with matched baselines remain limited.
Sequential switching is a dynamical mechanism in which different units transiently dominate in a reproducible order without convergence to a single permanent winner. Such behavior is often linked to saddle-type structures and is commonly generated through directed or explicitly asymmetric interactions. Here, we investigate whether recurrent sequential dominance can arise in a minimal discrete-time network with symmetric coupling. We study a three-unit network of non-chaotic Rulkov maps coupled through symmetric state-difference interactions. Although the coupling does not impose a preferred activation order, small intrinsic offsets in the excitability parameters break the degeneracy among the units and select reproducible switching sequences. We identify regimes displaying this behavior using simple criteria based on amplitude, exclusivity, recurrence, and order consistency. The dynamics are illustrated through representative trajectories and supported by local stability analysis near dominance configurations, which is consistent with saddle-type organization. Perturbation tests further show that sequential switching can reappear after transient forcing or persist under selected sustained drives. These results provide a concise dynamical characterization of sequential switching in a minimal symmetrically coupled non-chaotic Rulkov network.
The analysis of random walks on networks often relies on global quantities that average over nodes, thereby masking local differences in diffusion speed. This study introduces a vertex-level quantity Hi, defined as the finite-window fitted scaling exponent of the mean squared resistance distance ⟨Ωi2(t)⟩∼Cit2Hi from a given node i. We found nodes with Hi values below 0.5 (echo effect) and above 0.5 (catapult effect). The exponent is computed exactly via matrix powers of the transition matrix. We systematically evaluate Hi on several synthetic network families, generalized Sierpiński graphs, Newman-Watts small-world networks, and a custom grid-path-complete graph, and on two real-world networks (international E-road network and western U.S. power grid). We found nodes with Hi values less than 0.5 (subdiffusive regime) and greater than 0.5 (apparent superdiffusion) in both model networks and real-world networks. Analysis of model networks shows that when a node has an echo effect, its Hi value is less than 0.5, whereas when it has a catapult effect, its Hi value is greater than 0.5. In the two real networks, most nodes are in the subdiffusive regime and the overall heterogeneity of the local diffusion exponents is low, as indicated by Rényi indices of 0.0835 (E-road network) and 0.0555 (power grid). Comparisons with classical centrality measures indicate that Hi provides information not captured by those measures. The local diffusion exponent offers a vertex-level, dynamics-based tool for identifying structural bottlenecks and node roles, complementing global network characterizations.
The permutation Jensen-Shannon divergence has been shown to be a versatile tool of great value for diverse applications within the time-series analysis field. Here, we discuss the robustness of this ordinal measure by analyzing the finite-size bias of its estimated values. Synthetic data generated from several stochastic and chaotic systems were carefully analyzed as testbeds, and relevant conclusions were extracted. Finally, to demonstrate the implementation and utility of these findings in a practical context, we analyze the data generated from a complex real-world system of high interest today: Bitcoin prices.
Vaccination behavior is closely coupled with disease transmission through individual risk perception and age-dependent contact patterns. However, the combined effects of imperfect vaccine protection and age-specific mixing on behavior-epidemic dynamics remain insufficiently characterized. We develop a network-based dynamical model that couples susceptible-infected-recovered-vaccinated (SIRV) transmission with an evolutionary vaccination game. Empirical age-specific contact matrices are used to represent interactions among age groups and to examine how contact heterogeneity shapes vaccination behavior and disease transmission. Approximate theoretical analysis and stochastic simulations indicate that vaccination uptake depends on the balance between individual protection and population-level feedback. When vaccine efficacy is high, individual incentives are stronger and promote higher vaccination coverage. When efficacy is low, population-level feedback becomes more important for limiting free riding. Age-structured contacts further amplify inter-group differences, with highly connected age groups showing higher vaccination uptake and a stronger effect on epidemic outcomes. These results identify age-specific contact patterns as a key structural factor in behavior-transmission coupling. The proposed framework provides a basis for studying immunization strategies in structured populations.
Distinguishing data-generating processes from finite observations is a foundational problem in nonlinear time series analysis. Compression ratios offer a model-free diagnostic of sequential complexity, yet are typically computed at a single sample size, discarding the information contained in how the ratio converges. This paper treats the convergence curve itself as the primary observable: fitting compression ratios across prefix lengths to a three-parameter scaling law, R(N) = a + bN-c, yields three diagnostics from a single analysis. The asymptote a provides a rank-faithful entropy rate proxy (Spearman ρ = 0.977 across 36 processes with known analytical rates), performing on par with the best point-estimate measures. The convergence exponent c captures how quickly the compressor exhausts learnable structure-a dimension not captured by any of the seven benchmark complexity measures tested here, including effort-to-compress (maximum |r| = 0.34). Among generators sharing identical entropy rates, c still varies substantially (mean range 0.173 across 22 same-entropy clusters), distinguishing processes that every point-estimate method considers equivalent. The compression consistency (R2 of the scaling-law fit) serves as a compositional heterogeneity diagnostic: R2 degrades systematically when data from multiple generating processes are temporally concatenated, but not for independent and identically distributed mixtures of the same generators. Deterministic chaotic maps are flagged by Bayesian-information-criterion preference for the logarithmic convergence form motivated by Lempel-Ziv (LZ) asymptotics, manifesting as distinctively low c (<0.40) for generators in the chaotic regime under the power-law parameterization. The framework is validated on 152 synthetic generators spanning 25 families and 105 real-world datasets. Code and data are publicly available at https://github.com/generative-structure/cbad-finite-size-scaling.
We investigate the interaction characteristics of solitons on a spin-wave background with various nonlinear magnetic excitations in a Heisenberg ferromagnetic spin chain with twisting interactions. We show that interactions with spatially periodic Akhmediev breathers generate an additional phase contribution, which reconstructs the relative phase between the soliton and the spin-wave background and consequently induces structural deformation and state transitions. In contrast, interactions with spatially non-periodic Kuznetsov-Ma breathers, and rogue waves do not modify the relative phase and therefore preserve the soliton state. An explicit analytical expression for the additional phase is derived, and its magnitude is found to depend on the initial relative phase of the soliton, the number of magnons carried by the soliton excitation, and the spatial periodicity of the breather. Based on these parameters, phase diagrams are constructed to characterize relative-phase-controlled soliton-state transitions. These findings reveal the role of interaction-induced additional phases in governing magnetic soliton-state transitions and may be relevant for future magnonic information-processing schemes.
