Recent achievements in nonlinear dynamics, synchronization, and networks Free

The analysis of synchronization phenomena in the evolution of dynamical systems has been an active subject since the earlier days of physics, starting in the 17 $th$ century with Huygen’s findings that two very weakly coupled pendulum clocks hanging on the same wooden beam become synchronized in phase.¹ Later, phase synchronization phenomena were observed in chaotic oscillators by Rosenblum et al.² Other early examples are the synchronized flashing of fireflies or the behavior of adjacent organ pipes, which either reduce one another to silence or display synchronization.³ These phenomena are universal and can be understood within a common framework based on modern nonlinear dynamics. Nonlinear dynamics took a very active development in the early 1980s, and one of the key pioneers who propelled this field forward—and, in turn, deepened our understanding of complex synchronization phenomena—is Jürgen Kurths, whom we honor with this Focus Issue on the occasion of his 70 $th$ birthday. In his exceptionally productive career, Kurths has made seminal contributions to complex systems, nonlinear dynamics, and chaos theory. He is particularly noted for his discoveries of novel synchronization phenomena, recurrence, coherence resonance, measures of complexity, and causality, as well as dynamics and stability of complex networks, in all of which he has applied interdisciplinarily and introduced to various scientific disciplines (Fig. 1). What merits particular attention are his efforts to develop nonlinear dynamics, and thereby mechanistic understanding of phenomena across a diversity of fields, including solar and stellar activity phenomena; the Earth system; engineered, biological, and living systems; the climate; and other systems characterized by a high degree of complexity and nonlinearity (Fig. 2). The diversity of his research contributions and the impact of his research across a wide range of disciplines are reflected in this Focus Issue, which depicts the current progress and understanding across disciplines and fields.

Synchronization in coupled nonlinear oscillators is an emergent phenomena. In the beginning, this phenomenon was discovered between two coupled oscillators and was then extended to various types of networks with pairwise and non-pairwise interactions networks.^3,5 Recently, Messee Goulefack et al.⁶ show the impact of magnetic induction on the degree of coherence of a nonlocally coupled neuronal network. In the presence of induction, synchronization is more likely during resting periods, while in the active intervals, different levels of coherence can occur and spike-chimeras can emerge by varying strength of interneuronal coupling and quadratic memristor.

The study by Boaretto et al.⁷ explores phase synchronization in a network of randomly connected Hodgkin–Huxley neurons influenced by Poissonian spike trains through chemical synapses. They find that irregular spiking occurs within a specific range of external conductance when the coupling is off, while outside this range, the phase synchronization of neurons is reached and also an opposite effect, viz., the prevention of the network synchronization is observed for a very high value of the external currents. Furthermore, selective stimulation of neuron subsets shows that synchronization can extend to unstimulated neurons if the coupling is strong, but high external conductance disrupts this. Even stimulating a single neuron can induce network-wide synchronization with sufficient coupling. The study highlights the dependence of phase synchronization on the interplay between external and coupling currents, emphasizing the complexity of neuronal behavior, and here, the future work will be on the analytical approach to spiking activity and synchronization.

In addition to the advances in phase synchronization of neuronal networks, significant progress has also been made in the realm of multi-agent systems. Multi-agent systems have garnered significant attention over the past few decades due to their cost-effectiveness, increased robustness, and greater flexibility. Among these systems, the multiple rigid body system has seen growing interest because of its extensive applications in transportation, aerospace, and ocean exploration. However, synchronizing multiple rigid body systems is challenging due to the non-Euclidean configuration of attitudes and the inherent nonlinearity of their dynamics. The article by Jin et al.⁸ reviews recent progress in this area, focusing on attitude synchronization and coordinated motion control. It highlights key advancements, including local, global, and almost global attitude synchronization, as well as the integration of rotational and translational dynamics for coordinated control. The purpose of that study is to provide an overview of the latest developments in the synchronization of numerous rigid body systems from the perspectives of two basic issues: coordination control of various rigid body systems and attitude synchronization. The fundamental kinematic and dynamic paradigm for explaining rigid body systems is presented, along with significant findings and comparisons. Future research directions include addressing state constraints, achieving synchronization within a prescribed time, managing communication limitations, developing integrated frameworks for sensing, decision-making, and control, and improving experimental validation, especially in difficult environments.

Following the advances in multi-agent systems and synchronization of rigid body systems, significant progress has also been made in the synchronization of chaotic systems. Synchronization, where two coupled oscillators maintain a consistent dynamic relationship, was initially studied in periodic oscillators until Pecora and Carroll⁹ discovered it could also occur in chaotic systems. Then, it was shown that the relationship might be complex, leading to the empirical achievement of generalized synchronization in the absence of a systematic approach. With an emphasis on global observability and controllability, the work done by Letellier et al.¹⁰ investigates a novel method of achieving generalized synchronization through the use of a flat control rule developed from the nonlinear control theory. This technique entails positioning sensors and actuators in the best possible positions to ensure global observability and controllability. The type of synchronization obtained varies with the control gain. The study shows that the response system can achieve topological equivalency in its behaviors by dynamically mirroring the drive system with the right control parameters. Generalized synchronization can be preserved even with decreased control stiffness; however, higher-dimensional embedding of the response dynamics might be necessary. The study explores the implications for reservoir computing applications and emphasizes how flat control principles can be used to synchronize systems with a few sensors and actuators.

Following the exploration of generalized synchronization through flat control laws, recent research has also addressed the need to disrupt unwanted synchrony in coupled oscillators. Synchronization, though naturally occurring when oscillators interact, can be undesirable or harmful in certain situations. The paper by Mau and Rosenblum,¹¹ motivated by applications in neuroscience, focuses on a specific scenario involving two coupled oscillators where stimulation directly affects only one oscillator, and observation is limited to one unit, either the stimulated or the unstimulated one. The goal is to desynchronize the system using pulsatile stimulation, achieved by delivering pulses whenever the observed oscillator reaches a predetermined trigger phase. The authors develop a theoretical framework grounded in the assumptions of weakly coupled phase oscillators and brief pulses, describing the system’s dynamics through a dynamical map. This approach employs a phase-isostable description to establish a relationship between the response curves of individual oscillators and the coupled system. The study is supported by direct numerical simulations of a system with coupling functions that include higher harmonic terms. Optimization of this method involves selecting an appropriate trigger phase and adjusting the strength and polarity of the pulses. The research also details how to extract the necessary information from system observations and addresses the limitations of the proposed technique.

Further research has concentrated on improving synchronization in expanding networks of coupled oscillators, building on efforts to break undesired synchrony. Periodic motion is a common feature of dynamical systems, which behave in different ways depending on synchronized states. Since these systems frequently grow over time, methods for preserving or improving synchronization are required. To improve overall synchronization, the study by Park et al.¹² explores ways to optimize the natural frequency of newly added oscillators to an expanding network. In contrast to conventional techniques that employ a global order parameter, the researchers’ efficient growth schemes are based on a link-wise order parameter and, thus, exhibit better performance. The study demonstrates that these tactics lead to much better synchronization through numerical simulations and theoretical analysis. Beyond just performing as expected, the suggested strategy beats competing strategies consistently throughout a wide variety of coupling strengths. By giving useful advice for controlling synchronization in dynamically growing networks, this work sheds light on how oscillator systems evolve while maintaining their functional stability. The study shows that a linear combination of the natural frequencies of the current nodes is the ideal value for synchronization improvement, indicating an intrinsic node attribute connected to synchronization anti-affinity. This should be compared to the dynamical connectivity matrix in future studies. By applying the swing equation, the present method, which is based on the Kuramoto model in a linear interaction regime, might be investigated in different oscillatory systems, such as power-grid systems. This use case might confirm the methodology and reveal fresh directions for synchronizing expanding systems research. By approaching every node separately and independently, and then optimizing natural frequencies for improved synchronization, the techniques might also be used in fixed-size systems.

The study by Aravind and Meyer-Ortmanns¹³ investigates the relaxation times of heteroclinic motion after a sudden change in system parameters, focusing on the transition from oscillatory states to equilibrium. Slow relaxation indicates stability and memory retention, while fast relaxation aids in quick adaptation. Simulations show that slow relaxation is supported by a nested attractor space, especially across different bifurcations, but excessive noise can disrupt this. Sensitivity to initial conditions and noise varies, with single units being more affected than coupled systems. Coupled configurations reveal that relaxation depends on synchronization and the fraction of quenched units. The research suggests parallels between slow relaxation in complex systems and inertia in power grids, with potential applications in understanding brain dynamics and conditions like Parkinson’s disease, where controlling relaxation times could explain certain malfunctions.

Kostin et al.¹⁴ examine the significant impacts of asymmetry on the dynamics near the bifurcations of symmetric systems. Specifically, asymmetry stabilizes an asymmetric stationary partially synchronized state where one oscillator subgroup, slightly larger, becomes dominant, with its order parameter increasing proportionally to the square root of the asymmetry degree. This stabilization occurs even with a large frequency difference between the two distribution modes, and for infinitely large frequency differences, there is a range of moderate coupling strengths where this partially synchronized state is the only stable regime. For moderate frequency differences, this stabilization leads to a new type of bistability, where two stationary partially synchronized states coexist: one with a higher global order parameter and synchronization parity between subgroups and the other with a lower global order parameter and significant differences between subgroup order parameters. Additionally, asymmetry reduces the parameter region where bistability between incoherent and partially synchronized states is possible, disappearing completely at a critical asymmetry value, which varies for different types of asymmetry: population disparity, differing subgroup coupling strengths, different noise levels, and phase delays. These findings suggest that even minor advantages or disparities between oscillator subgroups can significantly influence global dynamics, impacting areas such as robust communication network design, control of complex dynamical systems, neuroscience, and ecological system analysis where competing synchronization forces are present.

In the work by Berner et al.,¹⁵ the authors investigate the synchronization transition in a heterogeneous system of all-to-all coupled phase oscillators with a mixed-mode interaction function, specifically incorporating both the first and second Fourier modes. Heterogeneity is introduced through a uniform distribution of the oscillators’ natural frequencies. Depending on the coupling parameters corresponding to each mode, the system experiences different transitions to synchrony. Two distinct scenarios of first-order transitions to full synchrony: one involving a cascade of first-order transitions through coexisting two-cluster phase-locked states and the other featuring a first-order transition from an anti-phase phase-locked state to an in-phase synchronous state are identified. This study reveals that the interaction between the first and second mode of the interaction function expands the range of possible transition scenarios, resulting in a variety of two-cluster phase-locked states. By extending the collective coordinate approach, the analytic conditions for the existence of these clustered phase-locked states in the observed transition scenarios are investigated.

The characteristic properties and stereotype activity patterns in the brain networks have held significant work of interest over the decades. During the processing of sensory, cognitive, or motor events in EEG signals, two important phenomena serve insightful importance, namely, event-related desynchronization (ERD) and event-related synchronization (ERS). Depending on external measurements, brain networks reflect different spatiotemporal patterns in different frequency domains. The study by Lefebvre and Hutt,¹⁶ includes noise-driven amplitude modulation of power enhancement, or power reduction in the $γ$-frequency domain $30$– $60$ Hz. The theory is extended in terms of the stochastic mean-field and supported via simulations. The results provide a good agreement with real data. In addition to verifying the properties of ERS and ERD, the finite size fluctuations relative to the network size are also investigated. In conclusion, this plays a significant role in future exploration of various correlations.

Expanding on the exploration of network synchronization, another innovative approach involves local Dirac synchronization.¹⁷ This method uses the Dirac operator, inspired by quantum mechanics and grounded in algebraic topology, to study the dynamics of coupled nodes and link signals on arbitrary networks. Topological signals, which include dynamical variables associated with nodes, links, and higher-dimensional simplices of simplicial complexes are gaining attention in fields such as brain research and data science. Local Dirac synchronization reveals that harmonic modes of the dynamics oscillate freely while other modes interact non-linearly, leading to a collectively synchronized state as the coupling constant increases. Characterized by discontinuous transitions and the emergence of a rhythmic coherent phase, this method shows how topology, particularly Betti numbers and harmonic modes, influences the dynamics of topological signals. The theoretical results, validated by extensive numerical simulations on various network types, highlight the potential role of topology in the onset of brain rhythms and open new perspectives on the interplay between network structure and dynamics. These findings provide fresh insights into the dynamics of linked topological signals on simplicial complexes and networks. These avenues of inquiry into the interaction between topology and dynamics hold promise for future study, including potential applications in the field of brain research.

Recent studies show that a first order phase transition is an ingredient of emerging states in pairwise Kuramoto network with higher-order interactions. This first-order synchronization can be suppressed if we apply adaptation in only triadic coupling and rather pushes the system toward a second-order synchronous phase transition.¹⁸ In this case, the form of bifurcation changes from sub-critical pitchfork to super-critical pitchfork due to the adaptation parameter.

Anwar and Ghosh¹⁹ studied neuronal synchronization in time-varying higher-order networks. Recently, the concept of time-varying nature between more than two nodes is introduced and gives a variety of emerging states.^20–22 It is observed that lower synaptic strength is required for synchrony if we compare it with static higher-order neuronal ensembles and temporal neuronal networks with just pairwise interactions. The enhancement of neuronal synchrony is more prominent in the case of fast switching of network topology. One of the important observations is that unlike for static simplicial complexes, neuronal synchronization can sometimes occur due to the sole influence of higher-order interactions in time-varying simplicial complexes (i.e., when the influence of pairwise interactions is not considered).

Rybalova et al.²³ investigate numerically how the spatiotemporal dynamics of ring networks of non locally coupled chaotic maps are affected by additive Gaussian noise and, particularly, on the probability of discovering chimera states. The logistic map, the modified Ricker map, and the Hénon map each describe a single node. The two-dimensional graphs show the spatiotemporal regimes for each noise-free network in the parameter plane “local dynamics parameter vs coupling strength” and the characteristics of the shift from incoherence to full synchronization as the coupling strength rises is investigated. The diagrams of dynamical regimes for all three networks show a coherent window inside the region with profile discontinuities, which have been found to substantially impact the likelihood of discovering chimera states in the presence of noise.

Research focusing on extreme events,²⁴ in the last decade or so, started using dynamical systems to create extreme-events-like scenarios and thereby to understand the underlying mechanisms of origin of such events and to suggest possible methods of prediction. The mechanism of extreme events in single oscillators^25,26 and coupled network dynamical systems^27–30 were investigated. Recently, Roy and Sinha³¹ probed the influence of coupling on extreme events in the membrane potential in a network of coupled model neurons, under various coupling forms and coupling strengths. Local diffusive coupling enhances the probability of occurrence of both temporal and spatial extreme events. However, global mean-field coupling suppresses extreme events. Under strong diffusive coupling, the return maps display some broad structured patterns with clusters around characteristic return times. Whereas under weak diffusive coupling and global coupling, inter-event intervals lose all perceptible correlations, and the greatest cluster emerges around small inter-event intervals. This indicates the high probability of a sequence of extreme events occurring in quick succession. In addition to this, non-local diffusive coupling yielded a large window where temporal extreme events occurred, but the spatial profile was synchronized.

Zhao et al.³² explored the role of noise in stochastic dynamical systems in the emergence of extreme events. Three different routes to extreme events under noise have been put forth: (i) under Lévy noise, (ii) while considering the stationary Gaussian process as a randomly varying excitation, and (iii) under Poisson white noise where extreme events occur as a result of the momentary jump between any general and a rare attractor. Under case (i), the authors have also studied the system’s proneness to produce extreme events with respect to varying stability and skewness index. When the stability index is less, extreme events with a heavy tail are observed whereas for higher stability index, no extreme events are found. On the other hand, the value of skewness index decides the direction of the heavy tail. All these results are demonstrated for a class of nonlinear Duffing oscillators describing the rolling motion of a ship in marine navigation.

Leo Kingston et al.³³ explored the Zeeman laser model and found the emergence of large intensity events (LIEs) as a response to changes in the pumping parameter. Three sources of instabilities were established as a reason for the raise of LIE, namely, (i) breakdown of quasiperiodicity to chaos followed by interior crisis, (ii) quasiperiodic intermittency, and (iii) Pomeau–Manneville intermittency. A common scenario of a discontinuous large expansion of the attractor of the laser model occurs during the appearance of hyperchaos for all three cases. The transition to hyperchaos and the appearance of LIE are concurrent and discontinuous for all three cases and hysteresis-free except in the case of PM intermittency when the transition shows a shift in the critical point with forward and backward integrations. In addition, the authors also noticed a shift in the transition points to hyperchaos against the parameter shifts in the presence of weak noise for all three cases; however, the fundamental feature of a discontinuous large expansion of the attractor with the origin of LIE remains unchanged.

Tiana-Alsina and Masoller³⁴ analyzed the event-coincidence analysis in order to characterize the synchronicity of optical spikes emitted by two mutually coupled lasers. Specifically, they have found that counting the spikes that occur almost simultaneously (within a short time interval between them) allows a precise quantification of the degree of synchronization of the spikes. For this, the authors have utilized two measures, namely, (i) asymmetric measure (⁠ $q$⁠) and (ii) symmetric measure (⁠ $Q$⁠). With $q$⁠, they find a gradual transition from a regime in which spiking of laser-2 trails behind that of laser-1 to a regime in which laser-2 follows laser-1. Next, measure $Q$ reveals that the synchronization level is, in general, high and decreases gradually when laser-2 looses its leadership role. This gradual decrease is followed by a sharp increase when laser-1 becomes the leading laser. These measures help in the quantification of the degree of synchronization and, also, to identify the leading laser and the lagging one.

Competition among homogeneous and heterogeneous species in the environment for existence, survival, and reproduction is a fundamental attribute of the Darwinian theory of evolution. To increase cooperative behavior among species, recently researchers have started to apply the evolutionary game theory in ecological systems to overview the dynamics in the abundance of strategically driven species, namely, the strategic species.³⁵ Finally, these mathematical frameworks become hybrid models, which provide insight into the effects regarding information and payoff through any evolutionary game dilemma, that effects directly in the persistence of ecological species, and named the eco-evolutionary dynamics.^36,37

The study by Bai et al.³⁸ addresses the distributed generalized Nash equilibrium (GNE) tracking problem in dynamic non-cooperative games, where cost and constraint functions vary over time and are revealed after decisions are made. The authors first propose a distributed inertial online game (D-IOG) algorithm using mirror descent, which tracks Nash equilibrium with low average regret through a time-varying communication graph. By carefully selecting the step size and inertial parameter, the algorithm guarantees that regrets increase sub-linearly, provided deviations from the Nash equilibrium also follow a sub-linear pattern. The study then extends to time-varying coupled constraints, presenting a modified D-IOG algorithm incorporating primal-dual and mirror descent methods for GNE tracking. This extension allows the derivation of upper bounds for regrets and constraint violations. Additionally, the effects of inertia and different information transmission modes are explored. The effectiveness of the proposed algorithms is validated through two simulation examples, which also analyze parameter effects and graph connectivity. Future research directions include addressing nonlinear GNE problems, multi-objective optimization, and developing communication-efficient algorithms.

In another study by Majhi,³⁹ the impact of small committed minorities among cooperators and defectors on evolutionary game dynamics is investigated, where individuals are more likely to interact with those using the same strategy. It examines the synergy between similar interactions and social viscosity across three social dilemmas: the prisoner’s dilemma, the stag hunt, and the snowdrift game. The evolutionary scenario is theoretically explained using generalized replicator dynamics in a well-mixed population. Focusing on the roles of social viscosity (stubborn individuals with strong beliefs) and assortativity (preferential interactions among similar individuals), how these factors influence cooperation is explored. It looks at populations with both ordinary and zealous individuals across the three social dilemmas. The findings reveal that zealous cooperators enhance cooperation, while zealous defectors hinder it. Assortative interactions improve cooperation, even with defectors present. The study uses both computational and theoretical analysis with generalized replicator dynamics to identify critical curves distinguishing full and partial cooperation regimes due to social viscosity.

Jiang et al.⁴⁰ explain how including a punishment strategy improves cooperative chances in social dilemmas such as the public goods game. They investigate the effectiveness of costly punishment in promoting cooperation through human experiments. Their findings reveal that underestimating punishment risk plays a significant role initially, but with sufficiently high fines, this irrationality diminishes, and deterrence alone can protect the commons. However, high fines deter free riders and discourage some generous altruists. Consequently, the tragedy of the commons is mainly avoided by cooperators contributing their “fair share.” Additionally, larger groups need higher fines to achieve the desired prosocial effect. In conclusion, this study suggests that measures, such as imposing economic sanctions on non-cooperative countries may prove effective in climate negotiations. The findings from these collective-risk games demonstrate robustness across various group sizes and loss probabilities. With significant deterrence, all players tend to converge towards fair-share investment while maximizing the probability of achieving the target sum. For instance, in groups of size six with a loss probability of $1 2$⁠, only one out of ten groups reached the target, a result mirrored in this experiment under similar conditions.

The basic research in complex systems theory led by Jürgen Kurths allowed him to also achieve significant contributions to the understanding of biological systems, including cardio-respiratory systems, bone structural changes, neuronal activity dynamics, and gene regulatory networks.^41–50 Articles on dynamical properties of complex biological systems are, therefore, a major part of this special issue, broadening our understanding of the behavior of neuronal circuits in the brain, circuits of biochemical interactions among clock genes, and proteins that control our metabolism, dynamics of cardiac regulation, or human vocalization.

Following recent experimental evidence on large diversity among the constituents of these networks, i.e., single neurons, while still maintaining robust overall dynamics, Hutt et al.⁵¹ demonstrate that heterogeneity enhances network resilience to changes in network size and connection probability by quenching the system’s dynamic volatility.⁵¹ In a similar context, the study by Barać et al.⁵² shows that the most efficient strategy to enact collective failure in networks of coupled neuronal oscillators does not only non-monotonically depend on the coupling strength, but also on the distance from the bifurcation point to the oscillatory behavior of individual excitable units.⁵² Focusing on synchronization phenomena, Chen et al.⁵³ utilize models of the network of the cerebral cortex to demonstrate that the synchronization of the global brain network is a continuous transition, in contrast to an oscillatory transition by its local subnetworks in the absence of resource constraints. This sheds light on the trade-off between minimizing energy consumption and maximizing efficiency in neuronal networks. Further studying the dynamical properties of the brain cortex, Contreras et al.⁵⁴ demonstrate the independent coexistence between a synchronous oscillation rhythm and a transiently synchronous avalanche activity, based on non-local diffusive interactions and local dynamics in the vicinity of a canard transition between subthreshold and relaxation oscillations in a model of coupled FitzHugh–Nagumo units.

Extending a model of neuronal regulation of sleep-wake cycle with a model of the network of orexinergic neurons, Yao et al.⁵⁵ have not only provided a mechanistic description of previous experimental findings of co-regulation between these systems, but also discussed a physical mechanism underlying the frequent transitions observed in narcolepsy. Two other studies have utilized direct measurements to develop diagnostic models and tools for identifying arrhythmia type in patients (Bury et al.⁵⁶) or identifying patients suffering from obstructive sleep apnea from heart rate and beat-to-beat blood pressure recordings (Pilarczyk et al.⁵⁷). Utilizing model-based analysis of experimental optical-mapping recordings of cardiac electrical excitation of a canine ventricle, Marcotte et al.⁵⁸ discuss the need for new avenues for optimization of data assimilation schemes applied to cardiac electrical excitation, as the reconstruction of electrical excitation patterns through the unobserved depth of the tissue is essential to realizing the potential of computational models in cardiac medicine.

Going to biochemical models, Kennedy et al.⁵⁹ study the effect of a cell’s mechanical environment on the circadian rhythms by modifying a standard model of circadian oscillation using the experimental constraints and demonstrate that the activity of the mechanosensory molecules YAP/TAZ affect circadian rhythms by increasing the expression of the clock gene Bmal1. This further suggests that circadian rhythms only occur when those mechanosensors are in the right activity range, consistent with experimental findings. This physical models provide an important experimental framework to study biological systems is further demonstrated in the study of Inoue et al.,⁶⁰ focusing on the biomechanics of human vocalization. They demonstrate various cooperative behaviors between the vocal and ventricular folds which arises in this coupled system, demonstrate the underlying dynamical mechanism, and discuss how the interpretation of these results can be utilized for the diagnosis of voice pathologies, such as ventricular fold dysphonia.

With respect to natural, multi-agent systems, in the study by Pattanayak et al.,⁶¹ the authors have explored the mode distribution of the transient time of predator population in a bistable tri-trophic food chain model. The authors show that the number of modes and the mean transient time decrease with increased distance from the basin boundary. The structure of the basin in terms of the number of modes in transient time distribution by introducing two new metrics, i.e., the homogeneity index and local isotropic index, is characterized. Here, the qualitative trend of the distribution of transient time and characteristics of the basin topology are independent of the nature of the attractor (chaotic or periodic). The authors explained the origin of these multimodal distributions and attempt to present their ecological implications.

The study by Sar and Ghosh⁶² involves the multi-agent systems commonly found in nature such as in schools of fish, colonies of honey bees, flocks of birds, etc. Several important questions are addressed, including: how the agents move in space without colliding with each other, how their directions change, and under which circumstances they achieve flocking with different structure formations. A minimal mathematical model of a locally interacting many-body system in which the agents simultaneously swarm in space and exhibit flocking behavior is proposed. The authors have also found that the agents achieve directional synchrony⁶³ when the directional coupling strength exceeds a critical threshold, and also additionally considered the effects of noise which is typical for natural systems.

An important concern in dynamical systems is the stability of dynamics, especially when considering environmental or ecological systems, and their potential for abrupt changes. Thus, stability and bifurcation analysis of solutions of nonlinear systems has been the cornerstone of the study of complex dynamical systems. In this collection, there are a few studies which demonstrate rich bifurcation scenarios in various systems ranging from nonlinear circuits to realistic vehicle systems. Usually, a bifurcation is induced by a variation in parameters. However, there are also bifurcations without parameters where the solution changes due to initial conditions for fixed parameters. A period-doubling (Feigenbaum) bifurcation without parameter was demonstrated in a nonlinear circuit with memristor.⁶⁴ In the spatiotemporal system of the cubic complex Ginzburg–Landau equation, it was shown that a stable spatially localized solution, dissipative solitons, can be realized with a double balance: external driving forces are compensated by dissipation effects; in addition, there is also a balance between linear dispersion and the nonlinear refractive index.⁶⁵ In a realistic four-wheel-steering vehicle system with driver steering control, a degenerate Hopf bifurcation, called Bautin bifurcation, was present, which allows the prediction of vehicle behavior after the loss of linear stability.⁶⁶ In unidirectional rings of bistable Duffing oscillators, multistable coexisting rotating waves can propagate to induce rich multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased.⁶⁷ Heterodimensional dynamics, where orbits with different numbers of unstable directions are inside a chaotic set, can appear via a collision of a chaotic attractor and a chaotic repeller.⁶⁸ This transition can be revealed by determining bifurcations of the periodic orbits embedded in the attractor and the repeller in a chaotically driven circle map.

While most analysis considers local linear stability with respect to small perturbations to the system state, Zheng and Boers⁶⁹ used a different approach by estimating the exit times from the absorbing and reflecting boundaries of the state as a global resilience measure. They successfully employ this concept to quantify the impact of environmental fluctuations and specific precipitation regimes on the stability of the Amazon rainforest state and its potential shift to a savanna type.

In typical systems with a stable solution, it takes an infinite stability start time to reach the stable state. In other systems, such as bidirectional associative memory networks, fixed-time stability, namely, a finite stability start time, the time needed to realize the associative memory, is crucial for ensuring reliable and robust system performance. By simplifying the calculation of the minimum upper bound for stability start time and reducing the complexity of stability analysis through a novel judging condition, the research by Chen et al.⁷⁰ contributes to the advancement of stability assessment methods in neural networks. This is important for applications in optimization, associative memory, pattern recognition, and other fields where stable neural network behavior is essential for successful operation.

Synchronization is a fundamental concern in three-phase AC power systems. In another study, Ma et al.⁷¹ review the recent progress on the synchronization stability and multi-timescale properties of the renewable-dominated power system (RDPS), from nodes and network perspectives. The paper outlines the bulk dynamical behavior within each timescale, employing a generalized approach that focuses on slower dynamics remaining unchanged while faster dynamics are considered damped, omitting engineering intricacies. It emphasizes the significance of the phase-locked loop as a nonlinear controller across all timescale dynamics. The authors categorize the slowest timescale and identify key controllers associated with each type of renewable generator. The authors review that by integrating the node dynamical and quasi-steady network models, a large-scale nonlinear model of the RDPS encompassing DC voltage and electromechanical timescales can be developed. Their study confirms the availability of differential-algebraic equations, albeit in a distinct manner, even with the fastest current scale dynamics. The paper’s primary objective is to provide a foundational understanding of the stability mechanism in the RDPS, addressing a gap in systematic studies and ongoing debates within the electrical power engineering field.

The mathematical models and their stability and bifurcation analysis presented in this collection have enriched the understanding of nonlinear dynamical systems and have potential applications in various fields, such as morphological computing, nonlinear optics, fluid dynamics, ecological dynamics, power system, and network dynamics.

Application of techniques of nonlinear dynamics and complex systems are often realized through abstracted models of reality. The challenge for modelers is to make the most appropriate assumptions, keeping the most salient features while disregarding irrelevance: models should be as simple as possible, but not simpler. Of course, models, of one sort or another underpin all physical science. But there are instances where it is the art of choosing the right model which draws out the key insights. This is also particularly true in biological applications.

Electrical signaling and propagation in heart tissue is one such example of nonlinear dynamics in a complex system where the model must capture the most salient aspects of reality. Angelaki et al.⁷² present one such model of action potential propagation. Their model captures realistic dynamics of both normal and damaged cells and allows for an accurate understanding of these complex dynamics in a homogeneous medium. Zakharova and Semenov⁷³ have studied neural firing through the Hindmarsh–Rose model and a recent model of Semenov. They observed a resonant noise effect—spatially localized activity patterns are stabilized for an appropriate level of noise. This leads the way to hypothesize on the persistence of local coupling activity in systems of neurons.

The dynamics of infectious diseases is an important problem – and another classic example of models with spatial structure and non-locality. In a research by Gabrick et al.⁷⁴ a specific type of a SIS model for the syphilis disease was developed with fractal derivatives. This helps to better capture the complex dynamics of disease spreading, including memory effects, long-range correlations, and non-local behaviors. In cases of infections spreading, seasonality is a considerable factor. A detailed dynamical study of a susceptible–exposed–infectious–recovered–susceptible (SEIRS) model, taking into account the periodicity of the transmission rate and its time dependence, is performed in the study by Brugnago et al.⁷⁵ The authors profoundly examined how the different frequencies of a seasonality function influence the dynamics of the system. They discussed the multistability of the system, and a critical analysis of the largest Lyapunov exponent is also done to find out large chaotic regions. A pinnacle outcome of this study elucidates the coexistence of chaotic and periodic attractors under the same parametric configurations and points out the scenarios where the infections in the periodic regions are greater than those in the chaotic ones.

Giammarese et al.⁷⁶ built a network based spatial model of the Amazon to study connectivity of the climate system. While quite a different application, there are still connections. They showed that connectivity—spatial coherence—in the Amazon is becoming less local and more long-range. The networks that they built allow them to then simulate and probe the effect of perturbations to the system. Unsurprisingly, the increased long-range connections lead to fast perturbation. While the Amazon is a potentially interesting example, the network based modeling approach that they employ will have applications in other climate systems.

The climate model that allows a more general range of dependency structures than either just the shortest possible or the longest possible ranged behaviors while allowing both as limiting cases. In this connection, Hasselmann realized that Brownian motion could be used as a mathematical superstructure to organize fast weather and slow climate fluctuations and that was a very powerful one, and his recent Nobel prize provided a timely context for the complex systems field. Watkins et al.⁷⁷ proposed the use of such a formalism to extend Hasselman’s energy balance models, the stochastic Mori–Kubo generalized Langevin equation.

Among the machine learning frameworks, reinforcement learning has a unique advantage in that the interactive training process is aligned with human decision-making. Abrupt transitions, unpredictable hidden causal factors, and heavy tail parameters all contribute to make financial markets difficult to model. Recently, a heavy-tailed preserving normalizing flow model is proposed that employs invertible transformations to simulate the high dimensional joint probability of the trading environment.⁷⁸ The model is verified using different financial markets data.

Synthetic power grids have become an important tool for studying the dynamics of power systems. However, it is important to construct future power grids model correctly capture effects such as blackouts. Jürgen Kurths worked on power grid systems in the last few decades. Recently, a synthetic power grid model for studying collective dynamical effects is introduced.⁷⁹ The model is combined to obtain synthetic power grids: realistic grid topologies, active power set points, and short-term fluctuations, as well as node and line models. This result enables the transient analysis of the collective dynamics of future power grids and especially their transient stability.

Finally, stochastic models represent an important class of models. Kwasniok⁸⁰ discusses approaches to data-driven (or inverse) modeling using statistical techniques. This approach is particularly useful when the system’s equations of motion are unknown, or when the aim is to derive reduced or effective equations that model specific macroscopic variables or modes of the system.

Nonlinear time series analysis was the first field in which Jürgen Kurths became scientifically active.⁸¹ He successfully used tools like entropy, correlation dimension, and Lyapunov exponents to investigate solar radio emission events.^82,83 In his career, although broadening his interests and expanding his scientific research to other fields, he always was interested to further contribute to the field of nonlinear time series analysis, e.g., by supporting the further development of recurrence plot based methods,^84,85 using complex networks for time series analysis,^86,87 or applying nonlinear data analysis to research questions in many different disciplines, as in engineering, life sciences, or climate.^{43,84,88–91} Nonlinear time series analysis is still an important and lively field with many different directions, as presented by the collection of papers in this Focus Issue.

The quantitative description of nonlinear systems is usually based on the phase space representation of their dynamics, using Lyapunov exponents,⁹² correlation dimension,^93, $K 2$ entropy,⁹⁴ recurrence properties,⁸⁵ etc. Recent years have seen new developments and ideas to reconstruct the high-dimensional phase space from time series.^95,96 In this Focus Issue, Börner et al.⁹⁷ suggest a novel approach using a detection matrix and demonstrate its suitability for short and single time series. This method is robust and limited only by the temporal resolution of the time series. Two further studies of this Focus Issue use the recurrence plot approach to study such phase space representations.^98,99 Antary et al.⁹⁸ investigate the effect of irregular sampling and interpolation on the results of the recurrence analysis, a common problem when working with paleoclimate data. They further provide a correction scheme to reduce these interpolation effects. John Bejoy and Ambika⁹⁹ apply recurrence plot analysis on temperature and relative humidity to compare the dynamics of different climate zones in India. They find that climate variability has shifted to more irregular dynamics in the early 1970s but shifted back after 2000. A similar approach to recurrence is suggested by Bilbao et al.,¹⁰⁰ but using and comparing a relaxing definition of recurrences. The relationships of these recurrences are considered hyperedges of a hypergraph, which can be used to differentiate between different systems.

Apart from the analysis of the continuous state space of a dynamical system, the discretization of the state space to generate symbolic dynamics of a dynamical system is useful in nonlinear time series analysis because we can analyze time series quickly and rigorously. However, the partition of the state space may face challenges for dynamical systems with noise. Hirata and Amigó¹⁰¹ present a review of symbolic dynamics from the perspective of nonlinear time series analysis and explain how the combinatorial nature of ordinal patterns and recurrence plots makes it possible to extend this symbolic-dynamical approach from deterministic to random time series. A similar approach to characterize dynamical systems is based on combining symbolic dynamics with the complexity–entropy plane.¹⁰² Kottlarz and Parlitz¹⁰³ reconsider the approach using ordinal pattern statistics (permutation entropy) and statistical complexity and investigate high-dimensional chaotic systems. Their important finding is that high-dimensional systems can sometimes not be easily discerned from random systems. The common practice of visual inspection and evaluation of the complexity–entropy plane can, therefore, be misleading for short time series of high-dimensional systems.

Another application of information theory is demonstrated by Butail et al.,¹⁰⁴ who use nonlinear estimation and information theory on time-series data from the COVID-19 pandemic to identify drivers of infection rates, such as mobility, masking, and immunity loss. To uncover these hidden relationships, they integrate the unscented Kalman filter with a detailed SEIR model, allowing for a robust analysis of the influences on infection dynamics.

Complex networks have become a universal tool for studying key structural features, in terms of network metrics, and understanding the emergence of collective phenomena in various real-world networked systems. Jürgen Kurths is one of the pioneers in this field and has published many seminal papers.^{86,87,105,106} He first focused on the influence of topology on system dynamics and then applied this tool to multidisciplinary fields such as heartbeat, climate, and brain. For example, since 2008 he has focused on the application of complex networks to climate, particularly for the prediction of extreme climate events.^107–110 

For the study of structural features, Zamora-López and Gilson¹¹¹ show that many classic network metrics can be derived from common hidden propagation models because of the spatiotemporal nature of the network’s response to external stimuli. This suggests that there is potential in designing new network metrics that can represent essential properties of real-world systems of interest. Yan et al.¹¹² investigate the relationship between the synergy and the triangular interaction in the simplicial contagion model and show that such relationship can be bridged together through the joint degree distribution. This result enhances the understanding of complex contagions in networked systems.

For the study of the emergence of collective phenomena, the network-coupled Kuramoto model is a widely studied model due to its mathematical tractability. Arenas et al.¹¹³ consider the Kuramoto model for a finite system of $n$ coupled oscillators and analyze the stability and properties of equilibrium solutions. Considering further such coupled Kuramoto systems, but with a biharmonic coupling function, Arefev et al.¹¹⁴ develop a numerical procedure for searching for heteroclinic cycles, as a potential indicator of chaos, and find that the presence of symmetries in this system may lead to the emergence of chaotic dynamics. Inversely, from the observational data of the phenomena that occur, it is possible to infer the underlying structural properties of the system. Song et al.¹¹⁵ apply machine learning techniques to the Kuramoto model to infer network structures from chaotic patterns. These techniques can also identify the transition point and criticality of a hybrid synchronization transition. This research shows that machine learning techniques can help understand the collective behavior of complex systems.

Both the structural and dynamical properties are crucial for the emergence of global system dynamics, and they are interrelated. Considering the relationship of complex networks to extreme events, Wassmer et al.¹¹⁶ study the transportation network stability during natural disasters. They develop a method based on the gravity model of travel to assess the impact of natural hazards on transportation networks, finding significant disruptions and highlighting the need for targeted infrastructure reinforcement. In a network coupled ecological consumer–resource model with noise induced symmetry breaking due to stochastic birth and death rates, Narang et al.¹¹⁷ show that increasing connectivity can increase species’ diversity. Additionally, in spatial interdependent networks, Gross et al.¹¹⁸ examine the dynamics of cascading failures identifying different types of phase transitions and proposing a new fourth type of phase transition where microscopic interventions have macroscopic effects, with implications for experimental observation and practical applications.

The work of Contreras-Aso et al.¹¹⁹ addresses a basic question of network based modeling methods. Often these methods rely on indicators of the network structure. One such widely used measure is centrality, and in particular, the so-called PageRank centrality. The influence of PageRank on the structure and function of the modern Internet is pervasive; however, Contreras-Aso and co-workers ask whether it is possible to manipulate this measure. While it is possible to modify the PageRank through weight changes, this work points to inherent stability in this particular modeling tool.

Wang et al.¹²⁰ build a climate network-based method that reveals the distribution of climate patterns and causal direction, determining the strength of teleconnections, and highlighting sensitive regions using global daily surface air temperature data. Using data from the first 37 years (1948–1984) and the last 37 years (1985–2021), respectively, two layers of the global climate networks on a temporal scale are constructed. A comparative analysis of the teleconnection intensity characteristics and distribution pattern on both layers is discussed. It is observed that between 1948– 1984 and 1985–2021, the majority of the teleconnections in the Northern Hemisphere are gradually becoming weaker, while the ocean’s influence on land in the Southern Hemisphere is significantly growing, creating several “hot spot” climate-sensitive areas (such as Southeastern Australia, Southern South America, and South Africa). Additionally, the results of this investigation showed that the strength of teleconnections is growing over time, which may be associated with human activity, global warming, and climate change.

In the context of epidemic spreading processes, dynamic multiplex networks offer a more accurate representation compared to those on single-layered networks. In the article by Feng et al.,¹²¹ the authors examine a two-layered network, divided into an information transmission layer and a disease spreading layer. They investigate the influence of these nodes, which do not employ any defensive measures, on disease spreading by comparing the selection patterns of different nodes. Utilizing the micro-Markov chain approach, the authors are able to derive the threshold of disease spreading for the proposed epidemic model, which demonstrates that the awareness layer affects the threshold. Their research demonstrates that individuals with high centrality within the awareness layer play a significant role in inhibiting the transmission of infectious diseases. The results of this study enhance the understanding of disease spreading on a two-layer network, especially the role played by different nodes in the awareness layer. Based on their analyses, local authorities should promptly inform individuals with significant influence on the awareness layer, such as certain news media departments and individuals with high visibility in the network, as well as broadcast messages to individuals who are not online, in order to effectively mitigate outbreaks.

The efficient detection of causality in the complex network is one of the interesting research topics since the last few decades. In this respect, Yang et al.¹²² present a conditional cross-map-based technique, to bridge the gap between pairwise causal detection and causal network reconstruction for the cross-map-based techniques, with arising problems such as indirect dependencies and common drivers being well settled. The model adopts the conceptual framework of dynamical causality and a useful tool to analyze and understand complex dynamical systems based solely on observational time series data.

Jürgen Kurths has been active in connecting the nonlinear dynamics community with researchers interested in control related applications and time-delay systems. He was one the main organizers of the PhysCon conferences and also hosted the 2007 conference in Potsdam. As such, several papers in this Focus Issue describe the impact of time-delays and control inputs on the dynamics of coupled systems.

The basin of attraction is an important property of attractors in dynamical systems. Valle et al.¹²³ demonstrate that ResNet50 CNN is very effective in learning the basin of attraction of paradigmatic nonlinear dynamical systems such as the Duffin oscillator, Hénon-Heiles Hamiltonian system, and a magnetic pendulum. Kent et al.¹²⁴ use next-generation reservoir computing to control the Hénon chaotic map with a very small number (e.g., 10) of training points with tunable convergence rate. Machine learning applications often require components with dynamical units that are simple but can be morphed into versatile learning tasks; Goldmann et al.¹²⁵ show that the complex dynamics of self-delayed feedback systems in reservoir computing has the potential to increase the dimensionality of the reservoir response.

Lainscsek et al.¹²⁶ use a network-motif delay differential analysis to study invasive intracranial electroencephalographic data from epilepsy patients; the identified network motifs have the potential to enhance clinical practice with novel seizure classification systems and improve seizure lateralization. Muñoz et al.¹²⁷ show that cycle-like structures naturally occur in anharmonically perturbed nonlinear oscillators, and these models can be relevant in interpreting the 11 year sunspot activity cycle without considering external factors affecting the solar system. Birythmicity is an important dynamical phenomenon that can occur in various physical, chemical, and biological systems; Biswas et al.¹²⁸ propose an ON–OFF controller to steer the dynamics to monorhythmic and stationary states. The location of the most influential sites can facilitate the suppression of wild fluctuations in the network. Bomela et al.¹²⁹ show with experiments using electrochemical oscillator networks that the most influential node can relocate for weakly or strongly coupled networks without any rewiring; for example, in a social network, the leader could change without making new connections but instead by strengthening the existing connections throughout the network.

Spin-torque nano-oscillators (STNOs) are inescapably prominent candidates for the study of nanoscale nonlinear dynamics and microwave generators at the nanoscale. A meticulous investigation about how the field-like torque influences the synchronization of a one-dimensional array of STNOs being coupled themselves via self-emitted currents is done by Arun et al.¹³⁰ Here, the numerical experiment is performed by solving the associated Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation. The very specific observation of this study corroborates that, if the number of oscillators is low, only field-like torque plays a crucial role in complete synchronization. On the other hand, if the number of oscillators is large, then the field-like torque along with the external field direction is important for the onset of complete synchronization. This is illustrated by considering separate arrays of 2, 10, and 100 oscillators.

As was the case with many other cross-pollinations between different scientific fields, Jürgen Kurths is again helping to form bridges and bring interesting concepts and ideas from different communities together. In this collection of papers, the work by Bogatenko et al.,¹³¹ coauthored by Jürgen Kurths, is proof of this, where the authors bring a paradigmatic model from dynamical systems to machine learning (ML): They incorporate the FitzHugh–Nagumo model of biological neurons into artificial neural networks, demonstrating improved accuracy in tasks such as image recognition and suggesting a pathway for integrating biological realism into machine learning methods.

On the other hand, several papers investigate how traditional methods might be enhanced with ML approaches. Chauhan et al.¹³² show how reservoir computing (RC) can be effective in predicting high-dimensional spatiotemporal patterns, specifically in phase ordering dynamics of 2D binary systems, utilizing an echo-state network to process large numbers of state variables in a computationally very efficient way. Xia et al.¹³³ extend the RC methodology to neural delayed reservoir computing (ND-RC), which decouples memory capacity and nonlinearity and allowing for independent tuning and proves successful in predicting systems with long-term dependencies, such as the Mackey–Glass system. Also, using neural networks, Patel and Ott¹³⁴ investigate the capability of neural network models to extrapolate beyond their training data to predict the long-term behavior of non-stationary systems, revealing conditions under which such ML methods succeed or fail, and suggesting potential improvements by integrating ML with conventional, knowledge-based models.

Finally, some contributions combine ML with the complex network theory. Andreev et al.¹³⁵ study the interpretability of the results of a highly accurate approach for distinguishing between patients with and without major depressive disorder based on feature selection and classification via linear discriminant analysis of fMRI and the identification of key features in the functional connectivity networks. Nauck et al.,¹³⁶ on the other hand, use graph neural networks (GNNs) to predict the dynamic stability of power grids, showing that GNNs effectively reduce computational costs and accurately identify vulnerable nodes in power grids, which they validate on synthetic and real-world grid models.

In this Focus Issue, 83 papers on recent advances in nonlinear dynamics, time series analysis, synchronization, machine learning, and networks highlight the enduring legacy of Jürgen Kurths’ work. His contributions have significantly expanded our knowledge in these areas and continue to shape future research directions.

We extend our sincere appreciation to the contributions of all the authors who submitted their interesting works to this Special Issue on “Nonlinear dynamics, synchronization and networks: Dedicated to Jürgen Kurths’ 70 $th$ birthday for Chaos (NDSNJK23).” We are very much thankful to all the reviewers who helped us to ensure the quality of the selected papers. We acknowledge the NDA23 conference during March 15–17, 2023 and funding from the DFG for the conference [DFG Project No. MA 4759/19: International scientific conference: “Nonlinear Data Analysis and Modeling: Advances, Applications, Perspectives (NDA23),” Potsdam]. Finally, we express our gratitude to the editorial office for their timely guidance and consistent support throughout the publication process.

The authors have no conflicts to disclose.

Dibakar Ghosh: Validation (equal); Writing – original draft (equal). Norbert Marwan: Validation (equal); Writing – original draft (equal). Michael Small: Validation (equal); Writing – original draft (equal). Changsong Zhou: Validation (equal); Writing – original draft (equal). Jobst Heitzig: Validation (equal); Writing – original draft (equal). Aneta Koseska: Validation (equal); Writing – original draft (equal). Peng Ji: Validation (equal); Writing – original draft (equal). Istvan Z. Kiss: Validation (equal); Writing – original draft (equal).