Deterministic chaotic systems often exhibit highly organized structures in both phase space and parameter space. In this work, we investigate two-dimensional Hénon-like extensions of the discrete Gaussian and cubic maps incorporating a linear feedback channel. The Gaussian-based model admits a natural realization as a nonlinear electronic circuit consisting of a Gaussian-saturating amplifier coupled to a linear memory feedback loop. A detailed bifurcation analysis reveals that, while the classical one-dimensional Gaussian map is primarily organized by fold and flip bifurcations together with period-bubbling routes to chaos, its two-dimensional extension possesses a substantially richer dynamical structure. In particular, the system exhibits Neimark-Sacker bifurcations, quasiperiodic dynamics, Arnold tongues generated through frequency locking, shrimp-shaped periodic stability regions embedded within chaotic regimes, and extensive multistability. The observed Arnold tongues follow the classical Farey-tree hierarchy of rotation numbers and organize the resonance structure of the parameter space. Furthermore, coexistence of attractors is demonstrated through bistability between period-9 and period-14 oscillations for identical parameter values. The corresponding basins of attraction display strong intermingling, and the computed basin entropy (Sb = 0.6032) indicates fractal basin boundaries and pronounced sensitivity to initial conditions. In addition, although the one-dimensional cubic map already exhibits shrimp-shaped periodic windows, its two-dimensional extension reveals nested shrimp families together with resonance tongue structures, substantially enriching the global bifurcation landscape. These results show that low-dimensional nonlinear maps can generate highly organized parameter-space architectures in which periodic, quasiperiodic, and chaotic dynamics coexist through well-defined bifurcation mechanisms.
We develop a data-driven framework for approximating the first same-direction return time and the associated Poincaré map of chaotic flows. Learning these quantities is challenging because the return-time field may exhibit sharp localized variations, and nearby initial conditions may undergo qualitatively different pre-return excursions. We introduce a symbolic residual-learning strategy based on finite itineraries of transverse section crossings. A global neural network captures the dominant trend, and a classifier routes class-dependent residual corrections through soft or hard rules. Reflection symmetry is incorporated using a symmetry-aware input representation and symmetry-reduced symbolic classes, while the predicted return time is supplied as an auxiliary feature to the Poincaré-map model. Experiments on the Lorenz and Shimizu-Morioka systems show that the global configuration using a logarithmic return-time target and a symmetry-aware input representation improves accuracy, while symbolic residual correction further reduces errors, especially in high-gradient regions. The return-time input also provides useful information for learning the Poincaré map, although the resulting gains are system dependent. The results reveal a qualitative correspondence between prominent return-time gradients and finite symbolic-class boundaries, supporting symbolic decomposition as a structured approach to learning chaotic return dynamics.
We investigate the impact of higher-order, or simplicial, interactions on the mean-field dynamics of a susceptible-exposed-infectious-quarantined-susceptible epidemic model on networks, under a degree-homogeneous mean-field approximation. For the corresponding mean-field system, we derive explicit conditions for the existence and local asymptotic stability of disease-free and endemic equilibria, and obtain closed-form expressions for the critical parameters β0, βc, and vc that govern the onset of subthreshold bistability. The analysis shows that the simplicial transmission coefficient does not alter the linear epidemic threshold, but it can create a bistable regime below βc in which the disease-free equilibrium coexists with two endemic equilibria. We then perform a numerical bifurcation and basin analysis combining equilibrium branches, phase diagrams, time-domain simulations, and basin-of-attraction plots. These results reveal how higher-order interactions enlarge the endemic equilibrium and induce a nontrivial separatrix between extinction and persistence. Finally, a Routh-Hurwitz analysis excludes a Hopf bifurcation of the larger endemic equilibrium under the baseline parameters, while numerical scans over wider parameter ranges provide additional support for the absence of Hopf instability in the tested regimes. The findings suggest that simplicial interactions enrich epidemic dynamics primarily through multistability and basin geometry rather than sustained oscillations, although oscillatory behavior in substantially different regimes remains possible.
We investigate the interplay between gain-loss dynamics and nonlinearity in the PT-symmetric nonlinear Dirac equation with a scalar-scalar power-law interaction (|Ψ¯Ψ|kΨ,k>0). Exact solitary wave solutions reveal that the PT-transition point is determined solely by the existence condition and is independent of the nonlinearity exponent k. Despite the presence of gain and loss, the energy remains conserved. Although charge and canonical momentum are not conserved, a PT-symmetric modification of the canonical momentum yields a new conserved quantity. This modified conservation law gives rise to unusual kinematic properties, including nonzero momentum for stationary solutions and zero-momentum states for moving solutions at specific values of the gain-loss parameter. Finally, linear stability analysis and numerical simulations show that gain-loss effects and higher-order nonlinearities reduce the stability region of solitary waves: Solutions remain stable for k ≤ 2 and become unstable in part of the parameter space for k > 2.
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.
