This editorial provides an overview of the Focus Issue on "Nonlinear Dynamics of Reservoir Computing: Theory, Realization and Application" published in Chaos: An Interdisciplinary Journal of Nonlinear Science. We highlight the diverse contributions that bridge the gap between the fundamental theory of dynamical systems and the practical implementation of reservoir computing. The collection explores novel theoretical frameworks, innovative hardware substrates, and cutting-edge applications in forecasting, denoising, and control.

In this study, a complex network rapidly switches between its restricted complement and itself while maintaining the network’s total weight. We show that by varying the switching duty cycle, the required coupling strength for network synchronization can be controlled and therefore optimized. The optimized coupling strength is smaller than either that of the original network and restricted complement. The analytical results are validated by simulating small and large random and scale-free (preferential attachment) networks of chaotic Hindmarsh–Rose neurons with 1–1 electrical coupling. Since the switching occurs at high frequency, the average topology can be considered, and the maximum conditional Lyapunov exponent of its variational equations and the synchronization error are calculated, serving as the synchronization criteria. The numerical results agree with the analytical basis and with each other. This study shows that rapid switching between the network and its restricted complement can create an effective average topology without requiring additional weight supplies, thereby controlling the system’s dynamic behavior at a layer between the two networks’ connection distributions. This is especially important for analyzing non-stationary networks and for designing structural control mechanisms.

Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behavior of a given epidemic or information diffusion process. A common approach to imbue more realism to the spreading process is to constrain simulations to a network structure, where connected nodes update their disease state based on pairwise interactions along the edges of their local neighborhood. Simplicial contagion models (SCM) extend this to hypergraphs such that groups of three nodes are able to interact and propagate the disease along higher-order hyperedges (triangles). Though more flexible, it is not clear the extent to which the inclusion of these higher-order interactions results in dynamics that are characteristically different from those attained from simpler pairwise interactions. Here, we propose an agent-based model that unifies the classical SIS/SIR compartmental model and SCM, and extends it to allow for interactions along hyperedges of arbitrary order. Using this model, we demonstrate how the steady-state dynamics of pairwise interactions can be made to replicate those of simulations that include higher-order topologies by linearly scaling disease parameters based on a proposed measure of network activity. By allowing disease parameters to dynamically vary over time, lower-order pairwise interactions can be made to closely replicate both the transient and steady-state dynamics of higher-order simulations. We demonstrate that this relationship is robust to misspecification in the assumed higher-order interaction model, and applies to non-clique complex hypergraphs with nontrivial heterogeneous topology. For the latter case, it is found that heterogeneities in hypergraph topology result in weakened approximations of higher-order dynamics by pairwise interactions.

The readout-only training mechanism establishes reservoir computing (RC) as a prominent lightweight prediction model, but simultaneously compromises its ability to effectively capture the specific higher-order interaction inherent to complex dynamical systems. To address this issue, a novel RC input layer inspired by simplicial complexes is proposed. As the direct interface to the input time series, this input layer acts as a potential multi-order feature extractor for explicitly and directly modeling the complex interactions within dynamical systems. Specifically, the novel input layer is initialized as a set of random simplices with varying dimensions, each of which is responsible for representing the interaction features of the corresponding order. Like its original counterpart, the presented input layer requires no training, thereby fully preserving the hallmark low training cost of RC. Furthermore, a causality-based quantification method is developed to measure the multi-order information richness of RC. Numerical experiments are then conducted to systematically analyze how the simplex distribution in the new input layer and key RC hyperparameters affect the quantified richness metrics. Finally, the proposed input layer can be extended to various RC variants, and its effectiveness in enhancing RCs’ prediction performance is validated through prediction tasks involving both chaotic systems and real-world datasets.

We derive a penalty strength criterion for ridge regression using stochastic complexity, which is a refined variant of the minimum description length principle. Since stochastic complexity does not typically account for the effect of regularization on complexity, despite its ability to simplify models, we are required to make a slight modification to the underlying coding scheme. Our scheme makes use of a weighted ensemble of regularized model fits rather than a mixture of maximum likelihood estimates. Under this modification, regularization is interpreted as reducing model complexity by constraining flexibility. In the case of ridge regression, the complexity penalty term that we derive can be expressed analytically as the log determinant of the residual operator. We demonstrate the effect of this complexity penalty by fitting a linear readout to a reservoir computer, and by performing benchmark testing on publicly available datasets.

Misinformation spreading on social media is generally restricted by a variety of factors such as the underlying network properties and user behaviors. We consider a part of users have skeptical or critical attitudes to misinformation. We propose a novel ignorant-hesitator-spreader-recovery (IHSR) spreading model on time-varying networks with a hesitation mechanism characterized by the proportion of skeptical nodes (users) and their skepticism level. Using a mean-field approach and Monte Carlo simulations, we verify the correctness of our model and investigate how the hesitation mechanism suppresses misinformation spreading, in terms of spreading threshold and the final prevalence of misinformation. It is shown that with the increase of the proportion of skeptical users and the increase of the skepticism level, the spreading threshold becomes greater and the final prevalence is significantly reduced. We also compare the impacts of three selection strategies of skeptical nodes on the final prevalence. Interestingly, a counterintuitive result is obtained that prioritizing selecting nodes with a smaller activity works best for suppressing misinformation, and by contrast prioritizing selecting nodes with a larger activity is worst.

We study resilience in collective behaviors of "next-generation reservoir computers" in terms of transmitted signal distortion. Specifically, we introduce an interactive communication scheme and achieve synchronization between two "next-generation reservoir computer" oscillators. A dynamical transition from synchronization to desynchronization emerges with the growth of signal distortion. Remarkably, we show that the order of clique has no significant effect on the robustness of synchronization. The effectiveness of our proposed scheme is illustrated via the classical dynamical models and qualitative analysis. Our work reveals the function of transmitted signal distortion in shaping collective behaviors of machine learning oscillators.

Uncovering the underlying complex network structure with high-order topologies from observational data is a fundamental challenge across diverse domains. Can we provide a versatile and precise approach for inferring full-order structure solely from node states? To address this issue, we propose a data-driven likelihood optimization framework for reconstructing the underlying topological structures. Our approach captures state transition relationships in binary time series generated by Markovian dynamics and employs the difference of convex algorithm to efficiently handle the optimization problem of full-order reconstruction. Experiments demonstrate that our approach excels in reconstructing a full-order simplicial network and achieves high accuracy across different experimental conditions, highlighting its potential to uncover complex interactions.

Traditional complexity measurement commonly emphasizes pattern structure and density estimation. Although the emergence of patterns inherently stems from the intrinsic autocorrelation and inter-correlations within signals, complexity measurements do not specifically cope with these correlations, leading to limited performance on multivariate time series. To address this limitation, we introduce time-delay embedding and singular value decomposition to permutation entropy (denoted as HES), which comprehensively characterizes the complexity of multichannel signals from multiple perspectives while reducing the impact of correlations. Our method achieves improved classification accuracy compared to state-of-the-art complexity metrics, and exhibits enhanced sensitivity to intrinsic mode memory changes. Furthermore, we combine HES with random forest or support vector machine classifiers and a surrogate optimization algorithm on three benchmark datasets. This combination method achieves higher accuracy which smoothly varies with embedding parameters. Moreover, HES excels in binary classification problem and is highly effective for short vibration signals.

The aim of this text is to provide a linguistically accessible, but comprehensive introduction into a variety of topics in dynamical systems and its applications. Whilst preliminary knowledge of dynamical systems is useful, it is not essential and readers are only assumed to have familiarity with foundational undergraduate mathematics topics of calculus, linear algebra and rudimentary statistics. A variety of extended topics on recent publications and research activities in the field have been included in the last four chapters, which the interested reader may use as an introduction into further reading. A collection of exercises and questions both theoretical and computational are also included in this text.

Uncovering underlying topological structures is fundamental to understanding complex systems, but it presents a massive combinatorial challenge that is further complicated when spreading processes exhibit non-Markovian, memory-dependent characteristics and involve the reconstruction of higher-order structures. To simultaneously address both the structural and dynamical complexities, we propose a likelihood inference framework for the full-order reconstruction of simplicial complexes from general non-Markovian binary time series. We introduce a target-specific exposure clock and a Weibull memory kernel to explicitly model nonlinear aging effects in higher-order contagion. Methodologically, our central contribution is a relaxation strategy that overcomes the non-convex optimization difficulties typical of memory-dependent models, yielding a globally concave log-likelihood landscape. This formulation enables sparse optimization via fast iterative shrinkage-thresholding algorithm to separate true simplices from correlated spurious candidates, untangling the complex temporal-topological coupling. Numerical experiments on synthetic and empirical networks demonstrate that the framework adapts to different memory effects and aging regimes, achieving robust reconstruction beyond the regime covered by conventional Markovian models, while also demonstrating its applicability when the true structural orders are unknown.

We use a robust methodology that enables us to detect synchronous regions in networks of coupled dynamical systems and identify their collective behaviors. Our method employs ordinal patterns of spatial configuration of neighbor oscillators at each time point to ascertain whether or not neighboring nodes in a network are synchronized. We then use permutation entropy and forbidden sequence cardinality to classify collective behavior. We first demonstrate the effectiveness of our method on a time series of coupled identical logistic maps that are located on a ring. Our method not only confirms previous findings of collective behavior identification but also shows the borders of synchronous regions when oscillators of a network are partially synchronized. Then we apply our findings to a network of logistic maps with random connections to demonstrate the method's efficacy in situations where the network's spatiotemporal plots are not feasible.

Community detection plays a crucial role in understanding the structural organization of complex networks. Previous methods, particularly those from statistical physics, primarily focus on the analysis of mesoscopic network structures and often struggle to integrate fine-grained node similarities. To address this limitation, we propose a low-complexity framework that integrates machine learning to embed micro-level node-pair similarities into mesoscopic community structures. By leveraging ensemble learning models, our approach enhances both structural coherence and detection accuracy. Experimental evaluations on artificial and real-world networks demonstrate that our framework consistently outperforms conventional methods, as well as state-of-the-art embedding-based and learning-based approaches, achieving higher modularity and improved accuracy in normalized mutual information and adjusted rand index. Notably, even in the complete absence of ground-truth community information, our approach still achieves substantial improvements in algorithmic accuracy based on the principles of statistical-physics methods. When ground-truth labels are available, it yields the most accurate detection results, effectively recovering real-world community structures while minimizing misclassifications. To further explain the performance of our framework, we analyze the correlation between node-pair similarity and evaluation metrics. The results reveal a strong and statistically significant correlation, underscoring the critical role of node-pair similarity in enhancing detection accuracy. Overall, our findings highlight the synergy between machine learning and statistical physics, demonstrating how machine learning techniques can enhance network analysis and uncover complex structural patterns.

This study introduces a comprehensive framework that situates information cascades within the domain of higher-order interactions, utilizing a double-threshold hypergraph model. We propose that individuals (nodes) gain awareness of information through each communication channel (hyperedge) once the number of information adopters surpasses a threshold phi m. However, actual adoption of the information only occurs when the cumulative influence across all communication channels exceeds a second threshold, phi k. We analytically derive the cascade condition for both the case of a single seed node using percolation methods and the case of any seed size employing mean-field approximation. Our findings underscore that when considering the fractional seed size, r0 is an element of (0, 1], the connectivity pattern of the random hypergraph, characterized by the hyperdegree, k, and cardinality, m, distributions, exerts an asymmetric impact on the global cascade boundary. This asymmetry manifests in the observed differences in the boundaries of the global cascade within the (phi m, (m)) and (phi k,(k)) planes. However, as r0 -> 0, this asymmetric effect gradually diminishes. Overall, by elucidating the mechanisms driving information cascades within a broader context of higher-order interactions, our research contributes to theoretical advancements in complex systems theory.

Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behaviour of a given epidemic or information diffusion process. A common approach to imbue more realism to the spreading process is to constrain simulations to a network structure, where connected nodes update their disease state based on pairwise interactions along the edges of their local neighbourhood. Simplicial contagion models (SCM) extend this to hypergraphs such that groups of three nodes are able to interact and propagate the disease along higher-order hyperedges (triangles). Though more flexible, it is not clear the extent to which the inclusion of these higher-order interactions result in dynamics that are characteristically different to those attained from simpler pairwise interactions. Here, we propose an agent-based model that unifies the classical SIS/SIR compartmental model and SCM, and extends it to allow for interactions along hyperedges of arbitrary order. Using this model, we demonstrate how the steady-state dynamics of pairwise interactions can be made to replicate those of simulations that include higher-order topologies by linearly scaling disease parameters based on a proposed measure network activity. By allowing disease parameters to dynamically vary over time, lower-order pairwise interactions can be made to closely replicate both the transient and steady-state dynamics of higher-order simulations.

While the assumption that dynamical systems are stationary is common for modeling purposes, in reality, this is rarely the case. Rather, these systems can change over time, a phenomenon referred to as concept drift in the modeling community. While there exist numerous statistics-based methods for concept drift detection on stochastic processes, approaches leveraging nonlinear time series analysis (NTSA) are rarer but seeing increased focus in cases where the processes are deterministic. In this work, we propose a novel approach to unsupervised concept drift detection in dynamical systems utilizing the embedding offered by a reservoir computing (RC) model. This approach is inspired by the performance of RC on supervised classification tasks that indicates a strong ability to characterize dynamical systems. We assess this method on a number of synthetic drifting data streams from dynamical systems as well as an experimental case concerning faulty ball bearing. Our results suggest that the RC based methods are able to generally outperform the existing NTSA methods across the test cases. We conclude our work with some comments regarding real-time implementation and the impact of hyper-parameters on the proposed algorithm.

We propose a universal method based on deep reinforcement learning (specifically, soft actor-critic) to control the chimera state in the coupled oscillators. The policy for control is learned by maximizing the expectation of the cumulative reward in the reinforcement learning framework. With the aid of the local order parameter, we design a class of reward functions for controlling the chimera state, specifically confining the spatial position of coherent and incoherent domains to any desired lateral position of oscillators. The proposed method is model-free, in contrast to the control schemes that require complete knowledge of the system equations. We test the method on the locally coupled Kuramoto oscillators and the nonlocally coupled FitzHugh-Nagumo model. Results show that the control is independent of initial conditions and coupling schemes. Not only the single-headed chimera, but also the multi-headed chimera and even the alternating chimera can be obtained by the method, and only the desired position needs to be changed. Beyond that, we discuss the influence of hyper-parameters, demonstrate the universality of the method to network sizes, and show that the proposed method can stabilize the drift of chimera and prevent its collapse in small networks.

This study presents a multiscale modeling framework that integrates stochastic integrate-and-fire dynamics, networked replicator dynamics, and a behavior-augmented susceptible-exposed-infectious-recovered-susceptible model to simulate the interplay between agent behaviors and infection spread. Traditional compartmental epidemiological models often overlook behavioral adaptations and network complexity that underlie disease transmission. In contrast, our approach captures the stochastic acquired viral load, reflecting how repeated exposures increase the risk of infection, while dynamically modeling the agent's protective behaviors as they adapt to perceived risks, imitation influence, and aggregate costs. The integration of these components allows the model to explicitly capture synchronization between behavioral adaptation and infection prevalence. Simulations reveal that this synchronized feedback can give rise to recurrent outbreaks, in which declining risk perception triggers a resurgence. More specifically, the integrate-and-fire mechanism buffers abrupt transitions, smoothing epidemic waves, while information delays exacerbate fluctuations. However, elevated risk awareness mitigates these disruptions by sustaining proactive behavior. By bridging evolutionary game theory, network science, and epidemiological modeling, this framework demonstrates that epidemic control depends not only on biological parameters but also on the timing, adaptation, and synchronization of collective behavior. Our findings underscore three critical components for epidemic control: timely communication, risk-aware policies, and feedback-responsive interventions. These strategies can stabilize disease dynamics and improve preparedness for future pandemics.