Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins

Explosive transitions in coupled Lorenz oscillators

We study the transition to synchronization in an ensemble of chaotic oscillators that are interacting on a star network. These oscillators possess an invariant symmetry and we study emergent behavior by introducing the timescale variations in the dynamics of the nodes and the hub. If the coupling preserves the symmetry, the ensemble exhibits consecutive explosive transitions, each one associated with a hysteresis. The first transition is the explosive synchronization from a desynchronized state to a synchronized state which occurs discontinuously with the formation of intermediate clusters. These clusters appear because of the driving-induced multistability and the resulting attractors exhibit intermittent synchrony (antisynchrony). The second transition is the explosive death that occurs as a result of stabilization of the stable fixed points. However, if the symmetry is not preserved, the system again makes a first-order transition from an oscillatory state to death, namely, an explosive death. These transitions are studied with the help of the master stability functions, Lyapunov exponents, and the stability analysis.

Master stability functions of networks of Izhikevich neurons

Synchronization has attracted interest in many areas where the systems under study can be described by complex networks. Among such areas is neuroscience, where it is hypothesized that synchronization plays a role in many functions and dysfunctions of the brain. We study the linear stability of synchronized states in networks of Izhikevich neurons using master stability functions (MSFs), and to accomplish that, we exploit the formalism of saltation matrices. Such a tool allows us to calculate the Lyapunov exponents of the MSF properly since the Izhikevich model displays a discontinuity within its spikes. We consider both electrical and chemical couplings as well as global and cluster synchronized states. The MSF calculations are compared with a measure of the synchronization error for simulated networks. We give special attention to the case of electric and chemical coupling, where a riddled basin of attraction makes the synchronized solution more sensitive to perturbations.

Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states

We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.

Learning unidirectional coupling using an echo-state network

Reservoir Computing has found many potential applications in the field of complex dynamics. In this article, we explore the exceptional capability of the echo-state network (ESN) model to make it learn a unidirectional coupling scheme from only a few time series data of the system. We show that, once trained with a few example dynamics of a drive-response system, the machine is able to predict the response system's dynamics for any driver signal with the same coupling. Only a few time series data of an A-B type drive-response system in training is sufficient for the ESN to learn the coupling scheme. After training, even if we replace drive system A with a different system C, the ESN can reproduce the dynamics of response system B using the dynamics of new drive system C only.

Intermingled attractors in an asymmetrically driven modified Chua oscillator

Understanding the asymptotic behavior of a dynamical system when system parameters are varied remains a key challenge in nonlinear dynamics. We explore the dynamics of a multistable dynamical system (the response) coupled unidirectionally to a chaotic drive. In the absence of coupling, the dynamics of the response system consists of simple attractors, namely, fixed points and periodic orbits, and there could be chaotic motion depending on system parameters. Importantly, the boundaries of the basins of attraction for these attractors are all smooth. When the drive is coupled to the response, the entire dynamics becomes chaotic: distinct multistable chaos and bistable chaos are observed. In both cases, we observe a mixture of synchronous and desynchronous states and a mixture of synchronous states only. The response system displays a much richer, complex dynamics. We describe and analyze the corresponding basins of attraction using the required criteria. Riddled and intermingled structures are revealed.

Controlling chimera states in chaotic oscillator ensembles through linear augmentation

In this work, we show how "chimera states," namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator ensemble, can be controlled through a linear augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through a master stability function. We find that these results are independent of a system's initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles.

A novel non-equilibrium memristor-based system with multi-wing attractors and multiple transient transitions

Based on the pure mathematical model of the memristor, this paper proposes a novel memristor-based chaotic system without equilibrium points. By selecting different parameters and initial conditions, the system shows extremely diverse forms of winglike attractors, such as period-1 to period-12 wings, chaotic single-wing, and chaotic double-wing attractors. It was found that the attractor basins with three different sets of parameters are interwoven in a complex manner within the relatively large (but not the entire) initial phase plane. This means that small perturbations in the initial conditions in the mixing region will lead to the production of hidden extreme multistability. At the same time, these sieve-shaped basins are confirmed by the uncertainty exponent. Additionally, in the case of fixed parameters, when different initial values are chosen, the system exhibits a variety of coexisting transient transition behaviors. These 14 were also where the same state transition from period 18 to period 18 was first discovered. The above dynamical behavior is analyzed in detail through time-domain waveforms, phase diagrams, attraction basin, bifurcation diagrams, and Lyapunov exponent spectrum . Finally, the circuit implementation based on the digital signal processor verifies the numerical simulation and theoretical analysis.

Riddled basins of attraction in systems exhibiting extreme events

Using a system of two FitzHugh-Nagumo units, we demonstrate the occurrence of riddled basins of attraction in delay-coupled systems as the coupling between the units is increased. We characterize riddled basins using the uncertainty exponent which is a measure of the dimensions of the basin boundary. Additionally, we show that the phase space can be partitioned into pure and mixed regions, where initial conditions in the pure regions certainly avoid the generation of extreme events, while initial conditions in the mixed region may or may not exhibit such events. This implies that any tiny perturbation of initial conditions in the mixed region could yield the emergence of extreme events because the latter state possesses a riddled basin of attraction.

Emergence of chimeras through induced multistability

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators-with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)-inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.