Driving-induced bistability in coupled chaotic attractors

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.

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.

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.

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

We study the multistability that results when a chaotic response system that has an invariant symmetry is driven by another chaotic oscillator. We observe that there is a transition from a desynchronized state to a situation of multistability. In the case considered, there are three coexisting attractors, two of which are synchronized and one is desynchronized. For large coupling, the asynchronous attractor disappears, leaving the system bistable. We study the basins of attraction of the system in the regime of multistability. The three attractor basins are interwoven in a complex manner, with extensive riddling within a sizeable region of (but not the entire) phase space. A quantitative characterization of the riddling behavior is made via the so-called uncertainty exponent, as well as by evaluating the scaling behavior of tongue-like structures emanating from the synchronization manifold.