Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling.

Cyclops states are intriguing cluster patterns observed in oscillator networks, including neuronal ensembles. The concept of cyclops states formed by two distinct, coherent clusters and a solitary oscillator was introduced by Munyaev et al. [Phys. Rev. Lett. 130, 107201 (2023)0031-900710.1103/PhysRevLett.130.107201], where we explored the surprising prevalence of such states in repulsive Kuramoto networks of rotators with higher-mode harmonics in the coupling. This paper extends our analysis to understand the mechanisms responsible for destroying the cyclops' states and inducing dynamical patterns called breathing and switching cyclops states. We first analytically study the existence and stability of cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional oscillators with inertia as a function of the second coupling harmonic. We then describe two bifurcation scenarios that give birth to breathing and switching cyclops states. We demonstrate that these states and their hybrids are prevalent across a wide coupling range and are robust against a relatively large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and switching cyclops states promise to strongly manifest in other physical and biological networks, including coupled theta neurons.

The hidden complexity of a double-scroll attractor: Analytic proofs from a piecewise-smooth system.

Double-scroll attractors are one of the pillars of modern chaos theory. However, rigorous computer-free analysis of their existence and global structure is often elusive. Here, we address this fundamental problem by constructing an analytically tractable piecewise-smooth system with a double-scroll attractor. We derive a Poincaré return map to prove the existence of the double-scroll attractor and explicitly characterize its global dynamical properties. In particular, we reveal a hidden set of countably many saddle orbits associated with infinite-period Smale horseshoes. These complex hyperbolic sets emerge from an ordered iterative process that yields sequential intersections between different horseshoes and their preimages. This novel distinctive feature differs from the classical Smale horseshoes, directly intersecting with their own preimages. Our global analysis suggests that the structure of the classical Chua attractor and other figure-eight attractors might be more complex than previously thought.

Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling.

Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops' eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications.

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

When multilayer links exchange their roles in synchronization.

Real world networks contain multiple layers of links whose interactions can lead to extraordinary collective dynamics, including synchronization. The fundamental problem of assessing how network topology controls synchronization in multilayer networks remains open due to serious limitations of the existing stability methods. Towards removing this obstacle, we propose an approximation method which significantly enhances the predictive power of the master stability function for stable synchronization in multilayer networks. For a class of saddle-focus oscillators, including Rössler and piecewise linear systems, our method reduces the complex stability analysis to simply solving a set of linear algebraic equations. Using the method, we analytically predict surprising effects due to multilayer coupling. In particular, we prove that two coupling layers-one of which would alone hamper synchronization and the other would foster it-reverse their roles when used in a multilayer network. We also analytically demonstrate that increasing the size of a globally coupled layer, that in isolation would induce stable synchronization, makes the multilayer network unsynchronizable.

Stability of rotatory solitary states in Kuramoto networks with inertia.

Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and oscillates with a different frequency. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (1) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (2) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.

Partial synchronization in the second-order Kuramoto model: An auxiliary system method.

Partial synchronization emerges in an oscillator network when the network splits into clusters of coherent and incoherent oscillators. Here, we analyze the stability of partial synchronization in the second-order finite-dimensional Kuramoto model of heterogeneous oscillators with inertia. Toward this goal, we develop an auxiliary system method that is based on the analysis of a two-dimensional piecewise-smooth system whose trajectories govern oscillating dynamics of phase differences between oscillators in the coherent cluster. Through a qualitative bifurcation analysis of the auxiliary system, we derive explicit bounds that relate the maximum natural frequency mismatch, inertia, and the network size that can support stable partial synchronization. In particular, we predict threshold-like stability loss of partial synchronization caused by increasing inertia. Our auxiliary system method is potentially applicable to cluster synchronization with multiple coherent clusters and more complex network topology.

Emergence of the London Millennium Bridge instability without synchronisation.

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. Yet, reviewing observational, experimental, and modelling evidence, we argue that increased coherence of pedestrians' foot placement is a consequence of, not a cause of the instability. Instead, uncorrelated pedestrians produce positive feedback, through negative damping on average, that can initiate significant lateral bridge vibration over a wide range of natural frequencies. We present a simple general formula that quantifies this effect, and illustrate it through simulation of three mathematical models, including one with strong propensity for synchronisation. Despite subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger. More broadly, we describe an alternative to Kuramoto theory for emergence of coherent oscillations in nature; collective contributions from incoherent agents need not cancel, but can provide positive feedback on average, leading to global limit-cycle motion.

Sliding homoclinic bifurcations in a Lorenz-type system: Analytic proofs.

Non-smooth systems can generate dynamics and bifurcations that are drastically different from their smooth counterparts. In this paper, we study such homoclinic bifurcations in a piecewise-smooth analytically tractable Lorenz-type system that was recently introduced by Belykh et al. [Chaos 29, 103108 (2019)]. Through a rigorous analysis, we demonstrate that the emergence of sliding motions leads to novel bifurcation scenarios in which bifurcations of unstable homoclinic orbits of a saddle can yield stable limit cycles. These bifurcations are in sharp contrast with their smooth analogs that can generate only unstable (saddle) dynamics. We construct a Poincaré return map that accounts for the presence of sliding motions, thereby rigorously characterizing sliding homoclinic bifurcations that destroy a chaotic Lorenz-type attractor. In particular, we derive an explicit scaling factor for period-doubling bifurcations associated with sliding multi-loop homoclinic orbits and the formation of a quasi-attractor. Our analytical results lay the foundation for the development of non-classical global bifurcation theory in non-smooth flow systems.

When three is a crowd: Chaos from clusters of Kuramoto oscillators with inertia.

Modeling cooperative dynamics using networks of phase oscillators is common practice for a wide spectrum of biological and technological networks, ranging from neuronal populations to power grids. In this paper we study the emergence of stable clusters of synchrony with complex intercluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. The populations have different sizes and can split into clusters where the oscillators synchronize within a cluster, but notably, there is a phase shift between the dynamics of the clusters. We extend our previous results on the bistability of synchronized clusters in a two-population network [I. V. Belykh et al., Chaos 26, 094822 (2016)CHAOEH1054-150010.1063/1.4961435] and demonstrate that the addition of a third population can induce chaotic intercluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd," which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the intercluster phase shifts can stably coexist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our stability results for predicting the emergence of chimeras and solitary states.

Synchronizability of directed networks: The power of non-existent ties.

The understanding of how synchronization in directed networks is influenced by structural changes in network topology is far from complete. While the addition of an edge always promotes synchronization in a wide class of undirected networks, this addition may impede synchronization in directed networks. In this paper, we develop the augmented graph stability method, which allows for explicitly connecting the stability of synchronization to changes in network topology. The transformation of a directed network into a symmetrized-and-augmented undirected network is the central component of this new method. This transformation is executed by symmetrizing and weighting the underlying connection graph and adding new undirected edges with consideration made for the mean degree imbalance of each pair of nodes. These new edges represent "non-existent ties" in the original directed network and often control the location of critical nodes whose directed connections can be altered to manipulate the stability of synchronization in a desired way. In particular, we show that the addition of small-world shortcuts to directed networks, which makes "non-existent ties" disappear, can worsen the synchronizability, thereby revealing a destructive role of small-world connections in directed networks. An extension of our method may open the door to studying synchronization in directed multilayer networks, which cannot be effectively handled by the eigenvalue-based methods.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results.

Chaotic attractors appear in various physical and biological models; however, rigorous proofs of their existence and bifurcations are rare. In this paper, we construct a simple piecewise-smooth model which switches between three three-dimensional linear systems that yield a singular hyperbolic attractor whose structure and bifurcations are similar to those of the celebrated Lorenz attractor. Due to integrability of the linear systems composing the model, we derive a Poincaré return map to rigorously prove the existence of the Lorenz-type attractor and explicitly characterize bifurcations that lead to its birth, structural changes, and disappearance. In particular, we analytically calculate a bifurcation curve explicit in the model's parameters that corresponds to the formation of homoclinic orbits of a saddle, often referred to as a "homoclinic butterfly." We explicitly indicate the system's parameters that yield a bifurcation of two heteroclinic orbits connecting the saddle fixed point and two symmetrical saddle periodic orbits that gives birth to the chaotic attractor as in the Lorenz system. These analytical tasks are out of reach for the original nonintegrable Lorenz system. Our approach to designing piecewise-smooth dynamical systems with a predefined chaotic attractor and exact solutions may open the door to the synthesis and rigorous analysis of hyperbolic attractors.

Overcoming network resilience to synchronization through non-fast stochastic broadcasting.

Stochastic broadcasting is an important and understudied paradigm for controlling networks. In this paper, we examine the feasibility of on-off broadcasting from a single reference node to induce synchronization in a target network with connections from the reference node that stochastically switch in time with an arbitrary switching period. Internal connections within the target network are static and promote the network's resilience to externally induced synchronization. Through rigorous mathematical analysis, we uncover a complex interplay between the network topology and the switching period of stochastic broadcasting, fostering or hindering synchronization to the reference node. We derive a criterion which reveals an explicit dependence of induced synchronization on the properties of the network (the Laplacian spectrum) and the switching process (strength of broadcasting, switching period, and switching probabilities). With coupled chaotic tent maps as our test-bed, we prove the emergence of "windows of opportunity" where only non-fast switching periods are favorable to synchronization. The size of these windows of opportunity is shaped by the Laplacian spectrum such that the switching period needs to be manipulated accordingly to induce synchronization. Surprisingly, only the zero and the largest eigenvalues of the Laplacian matrix control these windows of opportunities for tent maps within a wide parameter region.

When two wrongs make a right: synchronized neuronal bursting from combined electrical and inhibitory coupling.

Synchronized cortical activities in the central nervous systems of mammals are crucial for sensory perception, coordination and locomotory function. The neuronal mechanisms that generate synchronous synaptic inputs in the neocortex are far from being fully understood. In this paper, we study the emergence of synchronization in networks of bursting neurons as a highly non-trivial, combined effect of electrical and inhibitory connections. We report a counterintuitive find that combined electrical and inhibitory coupling can synergistically induce robust synchronization in a range of parameters where electrical coupling alone promotes anti-phase spiking and inhibition induces anti-phase bursting. We reveal the underlying mechanism, which uses a balance between hidden properties of electrical and inhibitory coupling to act together to synchronize neuronal bursting. We show that this balance is controlled by the duty cycle of the self-coupled system which governs the synchronized bursting rhythm.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

Foot force models of crowd dynamics on a wobbly bridge.

Modern pedestrian and suspension bridges are designed using industry standard packages, yet disastrous resonant vibrations are observed, necessitating multimillion dollar repairs. Recent examples include pedestrian-induced vibrations during the opening of the Solférino Bridge in Paris in 1999 and the increased bouncing of the Squibb Park Bridge in Brooklyn in 2014. The most prominent example of an unstable lively bridge is the London Millennium Bridge, which started wobbling as a result of pedestrian-bridge interactions. Pedestrian phase locking due to footstep phase adjustment is suspected to be the main cause of its large lateral vibrations; however, its role in the initiation of wobbling was debated. We develop foot force models of pedestrians' response to bridge motion and detailed, yet analytically tractable, models of crowd phase locking. We use biomechanically inspired models of crowd lateral movement to investigate to what degree pedestrian synchrony must be present for a bridge to wobble significantly and what is a critical crowd size. Our results can be used as a safety guideline for designing pedestrian bridges or limiting the maximum occupancy of an existing bridge. The pedestrian models can be used as "crash test dummies" when numerically probing a specific bridge design. This is particularly important because the U.S. code for designing pedestrian bridges does not contain explicit guidelines that account for the collective pedestrian behavior.

Bistable gaits and wobbling induced by pedestrian-bridge interactions.

Several modern footbridges around the world have experienced large lateral vibrations during crowd loading events. The onset of large-amplitude bridge wobbling has generally been attributed to crowd synchrony; although, its role in the initiation of wobbling has been challenged. To study the contribution of a single pedestrian into overall, possibly unsynchronized, crowd dynamics, we use a bio-mechanically inspired inverted pendulum model of human balance and analyze its bi-directional interaction with a lively bridge. We first derive analytical estimates on the frequency of pedestrian's lateral gait in the absence of bridge motion. Then, through theory and numerics, we demonstrate that pedestrian-bridge interactions can induce bistable lateral gaits such that switching between the gaits can initiate large-amplitude wobbling. We also analyze the role of stride frequency and the pedestrian's mass in hysteretic transitions between the two types of wobbling. Our results support a claim that the overall foot force of pedestrians walking out of phase can cause significant bridge vibrations.

Introduction: Collective dynamics of mechanical oscillators and beyond.

This focus issue presents a collection of research papers from a broad spectrum of topics related to the modeling, analysis, and control of mechanical oscillators and beyond. Examples covered in this focus issue range from bridges and mechanical pendula to self-organizing networks of dynamic agents, with application to robotics and animal grouping. This focus issue brings together applied mathematicians, physicists, and engineers to address open questions on various theoretical and experimental aspects of collective dynamics phenomena and their control.

Bistability of patterns of synchrony in Kuramoto oscillators with inertia.

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.

Synergistic effect of repulsive inhibition in synchronization of excitatory networks.

We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one's expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrate that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. This effect is generic and observed in different models of bursting neurons and fast synaptic interactions. We also find a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. This general law is in sharp contrast with linearly coupled networks with positive (attractive) and negative (repulsive) coupling where the placement and structure of negative connections heavily affect synchronization.

Spikes matter for phase-locked bursting in inhibitory neurons.

We show that inhibitory networks composed of two endogenously bursting neurons can robustly display several coexistent phase-locked states in addition to stable antiphase and in-phase bursting. This work complements and enhances our recent result [Jalil, Belykh, and Shilnikov, Phys. Rev. E 81, 045201(R) (2010)] that fast reciprocal inhibition can synchronize bursting neurons due to spike interactions. We reveal the role of spikes in generating multiple phase-locked states and demonstrate that this multistability is generic by analyzing diverse models of bursting networks with various fast inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, and the Purkinje neuron model.