Inferring the dynamics of a network of oscillators becomes a significant challenge in the absence of explicit system equations. We present a data-driven machine learning technique to predict different dynamical states of a network, specifically a star-structured one. The proposed method exploits a parameter-aware reservoir computing scheme based on the echo-state network (ESN) framework. Our method employs a minimal setup to learn the parameter-dependent dynamics of a large network, using only two ESN units. We utilize the topological symmetry of the network to reduce the training cost. We validate the performance of our scheme in both scenarios where the central node oscillator of the star network is identical and non-identical to the peripheral node oscillators. In both cases, the proposed scheme is able to efficiently predict various emergent multi-stable dynamics of the network with varied coupling strengths. Despite exposure to limited data during training, it shows notable performance in predicting unseen attractors, including chimera, coherent, incoherent, and cluster synchronization states present in the network dynamics. Thus, this study provides an efficient reservoir computing framework for learning the dynamics of large-scale oscillator networks.

Many complex systems with societal relevance are excitable, including brain tissue, cardiac tissue, heat waves, and epidemic spread. In cardiac tissue, arrhythmias often arise when electrical conduction is blocked by incomplete recovery after a previous stimulation. We recently presented a topological theory for excitable media that also captures conduction blocks. Therein, points where a wave front or a wave back spatially connects to a conduction block were shown to be topologically preserved, and denoted heads and tails, respectively. Here, we introduce algorithms to automatically localize heads and tails in excitation patterns on triangle meshes. We describe two variants, depending on the co-dimension of the forbidden zone in state space. As a result, the conduction block region is rendered either as a line of conduction block or an extended conduction block region. A key operation is to apply a bitwise OR operation onto states of a local vertex according to a partitioning into forbidden zone (Z), excited (E), unexcited (U), and their respective signed variants (S), which we abbreviate as ZEUS methods. The methods are applied to visualize heads, tails, and conduction blocks in simulated data and optical voltage mapping recordings of ventricular tachycardia in rabbit hearts. We compare the outcome to classical phase singularity analysis. Theoretical relations between the different options and advantages of each method are discussed. Robust algorithms are presented and made publicly available to identify certain topologically preserved points in excitation patterns. These methods can be used for automated analysis and classification.

Identifying the linear region interval is crucial for calculating correlation dimension (D2) and the largest Lyapunov exponent (LLE). In this paper, we propose a dynamic range identification method based on the single-objective programming theory. First, calculating the correlation integral and the average divergence exponent using the algorithm of Grassberger and Procaccia and the algorithm of Rosenstein et al.. Then, utilizing single-objective programming theory and taking the maximization of interval length as the objective function, while imposing constraints on the model's fitting accuracy, correlation between variables, etc., accurate identification of the linear region under the condition of optimal linearity is achieved. Finally, we used the least square method to fit the linear region to obtain D2 and the LLE, respectively. Some examples are used for numerical simulation to verify the effectiveness of our method. Compared to some mainstream methods, our method does not require a complex calculation process, and the results obtained are similar to theirs. In addition, this method provides a new idea for calculating the linear region.

We consider the basic tight-binding model for an array of waveguide arrays with periodic zigzag modulations in the longitudinal direction and local Kerr nonlinearity, focusing on the case with zero average modulation. From the Floquet spectrum of the linearized Su-Schrieffer-Heeger (SSH)-like system, we identify the various gaps where nonlinear solutions may exist, exponentially localized in the bulk and/or at edges. For the fully nonlinear system, numerical continuation yields families of exponentially localized Floquet lattice solitons, calculated to computer precision. Numerical Floquet linear stability analysis shows regimes of stability and explores instability scenarios appearing from internal mode resonances.

We study a generalized Kuramoto model in which each oscillator carries two coupled phase variables, representing a minimal swarmalator system. Assuming perfect correlation between the intrinsic frequencies associated with each phase variable, we identify a novel dynamic mode characterized by bounded oscillatory motion that breaks the π-reflection symmetry. This symmetry breaking enhances global coherence and gives rise to a non-trivial mixed state, marked by distinct degrees of ordering in each variable. Numerical simulations confirm our analytic predictions for the full phase diagram, including the nature of the transition. Our results reveal a fundamental mechanism through which detrained (dynamic) oscillators can promote global synchronization, offering broad insights into coupled dynamical systems beyond the classical Kuramoto paradigm.

We report the spontaneous alternation of rotating phase waves in a ring of reactively coupled nonlinear oscillators with heterogeneity. In contrast to uniform systems that support unidirectional phase rotation, heterogeneity can induce periodic alternation between clockwise and counterclockwise propagation. These transitions are mediated by transient standing-wave states and are accompanied by characteristic spectral sidebands indicating periodic alternation. Numerical integration and bifurcation analysis using AUTO suggest that alternation may arise from periodic dwelling near coexisting saddle-node bifurcation points, with the alternation frequency following a square-root scaling law with respect to heterogeneity strength. This mechanism highlights a deterministic route to symmetry selection in nonlinear media, providing a minimal model relevant to alternation phenomena in plasma, fluid, and biological systems. This system, thus, serves a minimal platform for studying symmetry-reselection transitions in extended nonlinear media.

We develop a minimal whole-heart model that describes cardiac electrical conduction and simulate a basic three-lead electrocardiogram (ECG). We compare our three-lead ECG model with clinical data from a Norwegian athlete database. The results demonstrate a strong correlation with the ECGs recorded for these athletes. We simulate various pathologies of the heart's electrical conduction system, including ventricular tachycardia, atrioventricular nodal reentrant tachycardia, accessory pathways, and ischemia-related arrhythmias, showing that the three-lead ECGs align with the clinical data. This minimal model serves as a computationally efficient digital twin of the heart.

Next Generation Reservoir Computing (NGRC) is a low-cost machine learning method for forecasting chaotic time series from data. Computational efficiency is crucial for scalable reservoir computing, requiring better strategies to reduce training cost. In this work, we uncover a connection between the numerical conditioning of the NGRC feature matrix-formed by polynomial evaluations on time-delay coordinates-and the long-term NGRC dynamics. We show that NGRC can be trained without regularization, reducing computational time. Our contributions are twofold. First, merging tools from numerical linear algebra and ergodic theory of dynamical systems, we systematically study how the feature matrix conditioning varies across hyperparameters. We demonstrate that the NGRC feature matrix tends to be ill-conditioned for short time lags, high-degree polynomials, and short length of training data. Second, we evaluate the impact of different numerical algorithms [Cholesky, singular value decomposition (SVD), and lower-upper decomposition] for solving the regularized least squares problem. Our results reveal that SVD-based training achieves accurate forecasts without regularization, being preferable when compared against the other algorithms.

This paper presents a unified framework for analyzing the input-output behavior of discrete-time complex networks viewed as open systems. Importantly, we focus on systems that are inherently modeled in discrete time-such as opinion dynamics, Markov chains, diffusion on networks, and population models-reflecting their natural formulation in many real-world contexts. By an open network, we mean one that is coupled to its environment, through both external signals that are received by designated input nodes and response signals that are released back into the environment via a separate set of output nodes. We develop a general framework for characterizing whether such networks amplify (pass) or suppress (block) the external inputs. Our approach combines the transfer function of the network with the discrete-time controllability Gramian, and uses the H2-norm as a comprehensive measure of signal gain across various classes of inputs. We introduce a computationally efficient network index based on the Gramian trace and eigenvalues, enabling scalable comparisons across network topologies. Application of our method to a broad set of empirical networks-spanning biological, technological, and ecological domains-uncovers consistent structural signatures associated with passing or blocking behavior. These findings shed light on how the network architecture and the particular selection of input and output nodes shape information flow in real-world systems, with broad implications for control, signal processing, and network design.

Understanding how external stimuli influence collective motion is essential for modeling biological systems and designing artificial swarms. Inspired by a recent experimental study conducted at Université Paris Cité, in which a school of fish responds collectively to a moving luminous stimulus, we develop a numerical model to reproduce these observed dynamics. The particles in our model are attracted to the light source, repelled from each other to avoid collisions, and subject to stochastic fluctuations modulated by the light intensity. By adjusting illuminance over time, our simulations successfully replicate key experimental findings, including the transition from disordered to ordered motion, the formation of cohesive groups, and the emergence of high polarization under strong light. We analyze polarization and average speed over time and demonstrate that the collective behavior undergoes a light-induced phase transition. Our results show strong agreement with experimental data and provide insight into the mechanisms governing collective response to dynamic environmental cues.

We consider a mortal random walker evolving with discrete time on a network, where transitions follow a degree-biased Markovian navigation strategy. The walker starts with a random initial budget T1∈N and must maintain a strictly positive budget to remain alive. Each step incurs a unit cost, decrementing the budget by one; the walker perishes (is ruined) upon depletion of the budget. However, when the walker reaches designated target nodes, the budget is renewed by an independent and identically distributed (IID) copy of its initial value. The degree bias is tuned to either favor or disfavor visits to these target nodes. Our model exhibits connections with stochastic resetting. The evolution of the budget can be interpreted as a deterministic drift on the integer line toward negative values, where the walker is intermittently reset to positive IID random positions and dies at the first hit of the origin. The first part of the paper focuses on the target-hitting statistics of an immortal Markovian walker. We analyze the target-hitting counting process (THCP) for an arbitrary set of target nodes. In the special case where a single target node coincides with the starting node, the THCP reduces to a renewal counting process. We establish connections with classical results from the literature. Within this framework, the second part of the paper addresses the dynamics of the evanescent walker. We derive analytical results for arbitrary configurations of target nodes, including the evanescent propagator matrix, the survival probability, the mean residence time on a set of nodes during the walker's lifetime, and the expected lifetime itself. Additionally, we compute the expected number of target hits (i.e., budget renewals) in a lifetime of the walker and related distributions. We explore both analytically and numerically a set of characteristic scenarios, including a forager scenario, in which frequent encounters with target nodes extend the walker's lifetime, and a detrimental scenario, where such encounters instead reduce it. Finally, we identify a neutral scenario in which frequent visits to target nodes have no effect on the walker's lifetime. Our analytical results are validated through random walk simulations.

The oscillatory dynamics of natural and man-made systems can be disrupted by their time-varying interactions, leading to oscillation quenching phenomena in which the oscillations are suppressed. We introduce a framework for analyzing, assessing, and controlling oscillation quenching using coupling functions. Specifically, by observing limit-cycle oscillators, we investigate the bifurcations and dynamical transitions induced by time-varying diffusive and periodic coupling functions. We studied the transitions between oscillation quenching states induced by the time-varying form of the coupling function while the coupling strength is kept invariant. The time-varying periodic coupling function allowed us to identify novel, non-trivial inhomogeneous states that have not been reported previously. Furthermore, by using dynamical Bayesian inference, we have also developed a Proportional Integral controller that maintains the oscillations and prevents oscillation quenching from occurring. In addition to the present implementation and its generalizations, the framework carries broader implications for identification and control of oscillation quenching in a wide range of systems subjected to time-varying interactions.

We investigate synchronization in a quantum reservoir computing (QRC) system when learning chaotic system of interest. By training a QRC model to learn the dynamical equations of chaotic systems, we confirmed its ability to capture the dynamics of nonlinear time series. Based on this, we constructed a drive-response synchronization framework consisting of two independently trained QRC models, and the response model was evaluated by analyzing the Euclidean distance between their predicted values. Additionally, we systematically study the influence of coupling strength on synchronization performance, revealing the crucial role of coupling parameters in the synchronization evolution. Moreover, this study not only demonstrated the potential of quantum reservoir computing in simulating chaotic systems but also verified the feasibility of synchronous prediction among multiple independent quantum reservoir systems under external driving by introducing a synchronization mechanism.

A mechanism whereby a speed-up or slow-down of thermalization or prethermalization of multimode photonic lattices (measured by the propagation distance required for the system's entropy to maximize) is proposed. The idea is to utilize the coexistence of extended and localized supermodes of lattices with mobility edge to prepare a wide range of input beams, which would cause the system to rapidly or slowly approach its equilibrium state. As a proof of concept, the generalized quasiperiodic Aubry-André potential is used to demonstrate such an accelerated-wind down "thermalization transition." The underlying ideas are also extended to photonic lattices that lack energy-dependent mobility edge in their spectrum. By altering the original system (whose spectrum corresponds to only locally confined or widespread supermodes), the coexistence of localized and extended states is achieved in the new structure of the eigenvalues-coupling constant diagram while keeping the total power unchanged and the Hamiltonian nearly the same. In essence, the perturbation changes the localization properties of some supermodes from being extended to confined and vice versa.

The microscopic structure of several amorphous substances often reveals complex patterns such as medium- or long-range order, spatial heterogeneity, and even local polycrystallinity. To capture all these features, models usually incorporate a refined description of the particle interaction that includes an ad hoc design of the inside of the system constituents and use temperature as a control parameter. We show that all these features can emerge from a minimal athermal two-dimensional model where particles interact isotropically by a double-well potential, which includes an excluded volume and a maximum coordination number. The rich variety of structural patterns shown by this simple geometrical model apply to a wide range of real systems including water, silicon, and different amorphous materials.

In this work, we propose a method for measuring the dynamical complexity of optical time-delayed (TD) chaotic systems. The mapping relationship between the system output and its time delay variant is learned by the reservoir computing (RC) network. The learning performance of RC networks can reflect the difficulty of the system's dynamics reconstruction and is quantified as a metric to measure the system complexity. For the well-known two kinds of optical TD chaotic generators, our metric appears more responsive to changes in system parameters compared to permutation entropy (i.e., a premier indicator used for complexity of optical chaotic time series), fractal dimension, and maximal Lyapunov exponent. We also study the relationship between the complexity metric and the time delay signature of optical TD chaotic systems. The results indicate that there is an inverse relationship between them over a wide range of parameters. We believe that this work can provide ideas for security evaluation of an optical chaotic generator from the perspective of underlying dynamics reconstruction.

There are numerous indicators used to characterize the degree of synchronization for a non-identical system consisting of heterogeneous phase oscillators, such as the critical coupling of phase synchronization and the critical coupling of frequency synchronization and order parameter. Is it possible to predict these indicators simultaneously given the realistic situations of unknown system dynamics, including network structure, local dynamics, and coupling functions? This process, known as multi-task learning, can be achieved through the model-free technique of a feed-forward neural network in machine learning. To elaborate, we can measure the synchronization indicators of a limited number of allocation schemes and utilize these data to train the machine model. Once trained, the model can be employed to predict these indicators simultaneously for any novel allocation scheme. More importantly, the trained machine can also identify the optimal allocation for synchronization from a large pool of candidates. This method solves an outstanding question, which is how to allocate a given set of heterogeneous oscillators on a complex network in order to improve the synchronization performance. Leveraging multi-task learning's ability to predict multiple synchronization indicators, we can ensure that the system with the optimal performs well throughout the entire synchronization transition. Additionally, we test the scalability of the machine; one approach is to predict the indicators for a system composed of a new set of oscillators, and the other is to simultaneously predict the indicators of different systems.

We study the classical planar two-center problem of a particle m subjected to harmonic-like interactions with two fixed centers. For convenient values of the dimensionless parameter of this problem, we use the averaging theory for showing analytically the existence of periodic orbits bifurcating from two of the three equilibrium points of the Hamiltonian system modeling this problem. Moreover, it is shown that the system is generically non-integrable in the sense of Liouville-Arnold. The analytical results are complemented by numerical computations of the Poincaré sections and Lyapunov exponents. Explicit periodic orbits bifurcating from the equilibrium points are presented as well.

The transmission lines we consider are constructed from the nonlinear inductors and the nonlinear capacitors. In the first part of the paper, we additionally include linear ohmic resistors. The dissipation thus being taken into account leads to the existence of shocks-the traveling waves with the different asymptotically constant values of the voltage (the capacitor charge) and the current before and after the front of the wave. For the particular values of ohmic resistances (corresponding to strong dissipation), we obtain the analytic solution for the profile of a shock wave. Both the charge on a capacitor and current through the inductor are obtained as the functions of the time and space coordinate. For the case of weak dissipation, we obtain the stationary dispersive shock waves. In the second part of the paper, we consider a nonlinear lossless transmission line. We formulate the simple wave approximation for such a transmission line, which decouples left- and right-going waves. The approximation can also be used for the lossy transmission line considered in the first part of the paper to describe the formation of the shock wave (but, of course, not the shock wave itself).

We present a investigation of solar active regions using a complexity-based framework that combines solar observations with methods from complex network theory. Building on the historical foundation that active regions constitute buoyantly emerging magnetic flux bundles, we leverage continuous multi-wavelength data, particularly synoptic magnetograms from SOHO, to track the morphological evolution and connectivity of these magnetically intense structures in the solar photosphere and low corona. We first identify active regions as topologically coherent features in the photospheric magnetic field, and subsequently construct graphs in which nodes represent individual or recurrent flux elements, while edges capture temporal adjacency. The resulting networks exhibit scale-free degree distributions, non-trivial clustering, and signatures of dynamic reconfiguration reminiscent of self-organized criticality. These emergent properties clarify how local flux emergence, reconnection processes, and coronal loop expansions collectively shape the global magnetic topology. In particular, we find that longer-lived active regions act as network "hubs," playing a critical role in the redistribution of magnetic energy. Our analysis reinforces the notion that solar magnetic fields evolve through multi-scale interactions, bridging global dynamo action with localized eruptions and shedding new light on the triggers of flares and coronal mass ejections. By uniting data-driven detection techniques with complexity-science tools, this work highlights how network representations can strengthen models of solar activity and refine our understanding of magnetic-field behavior across the solar interior and atmosphere.

Current research on trust game (TG) has revealed the interplay between individual strategies and external incentive mechanisms. However, how individual decisions are influenced by environmental factors and how dynamic incentive mechanisms can promote the evolution of trust remain open questions. In this paper, we use a dynamic margin mechanism to characterize environmental dynamics and establish the co-evolutionary dynamics of an asymmetric N-player TG with dynamic margins. The model consists of two components: the dynamics of the asymmetric N-player TG and the evolution of the margin coefficient. The former describes the trust dynamics between investors and trustees, while the latter represents the dynamic adjustment of the margin based on different combinations of strategies. We analyze the existence and stability of possible fixed points in the system dynamics and reveal the critical role of dynamic margins in shaping the evolutionary process. Furthermore, we introduce a penalty mechanism for non-investors and find that under specific conditions, an appropriate penalty can induce persistent oscillations in the system, thereby influencing the evolution of trust. Finally, numerical simulations further validate the theoretical analysis. In general, this work explores the role of dynamic margins in fostering collective trust from an environmental feedback perspective, providing new insights into the evolution of trust in complex social interactions.