Chittaranjan Hens

Extreme multistability: Attractor manipulation and robustness

The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.

Oscillation death in diffusively coupled oscillators by local repulsive link.

A death of oscillation is reported in a network of coupled synchronized oscillators in the presence of additional repulsive coupling. The repulsive link evolves as an averaging effect of mutual interaction between two neighboring oscillators due to a local fault and the number of repulsive links grows in time when the death scenario emerges. Analytical condition for oscillation death is derived for two coupled Landau-Stuart systems. Numerical results also confirm oscillation death in chaotic systems such as a Sprott system and the Rössler oscillator. We explore the effect in large networks of globally coupled oscillators and find that the number of repulsive links is always fewer than the size of the network.

Engineering generalized synchronization in chaotic oscillators

We report a method of engineering generalized synchronization (GS) in chaotic oscillators using an open-plus-closed-loop coupling strategy. The coupling is defined in terms of a transformation matrix that maps a chaotic driver onto a response oscillator where the elements of the matrix can be arbitrarily chosen, and thereby allows a precise control of the GS state. We elaborate the scheme with several examples of transformation matrices. The elements of the transformation matrix are chosen as constants, time varying function, state variables of the driver, and state variables of another chaotic oscillator. Numerical results of GS in mismatched Rössler oscillators as well as nonidentical oscillators such as Rössler and Chen oscillators are presented.

Chimeralike states in a network of oscillators under attractive and repulsive global coupling.

We report chimeralike states in an ensemble of oscillators using a type of global coupling consisting of two components: attractive and repulsive mean-field feedback. We identify the existence of two types of chimeralike states in a bistable Liénard system; in one type, both the coherent and the incoherent populations are in chaotic states (which we refer to as chaos-chaos chimeralike states) and, in another type, the incoherent population is in periodic state while the coherent population has irregular small oscillation. We find a metastable state in a parameter regime of the Liénard system where the coherent and noncoherent states migrate in time from one to another subpopulation. The relative size of the incoherent subpopulation, in the chimeralike states, remains almost stable with increasing size of the network. The generality of the coupling configuration in the origin of the chimeralike states is tested, using a second example of bistable system, the van der Pol-Duffing oscillator where the chimeralike states emerge as weakly chaotic in the coherent subpopulation and chaotic in the incoherent subpopulation. Furthermore, we apply the coupling, in a simplified form, to form a network of the chaotic Rössler system where both the noncoherent and the coherent subpopulations show chaotic dynamics.

Diverse routes of transition from amplitude to oscillation death in coupled oscillators under additional repulsive links.

We report the existence of diverse routes of transition from amplitude death to oscillation death in three different diffusively coupled systems, which are perturbed by a symmetry breaking repulsive coupling link. For limit-cycle systems the transition is through a pitchfork bifurcation, as has been noted before, but in chaotic systems it can be through a saddle-node or a transcritical bifurcation depending on the nature of the underlying dynamics of the individual systems. The diversity of the routes and their dependence on the complex dynamics of the coupled systems not only broadens our understanding of this important phenomenon but can lead to potentially new practical applications.

Transition from homogeneous to inhomogeneous steady states in oscillators under cyclic coupling

Abstract We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a negative parameter mismatch. Such a transition appears in limit cycle systems via pitchfork bifurcation as usual. Especially, when we focus on chaotic systems, the transition follows a transcritical bifurcation for cyclic coupling while it is a pitchfork bifurcation for the conventional diffusive coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle system and a Sprott system as a chaotic system. We verified our results analytically for cyclic coupling and numerically check all results including diffusive coupling for both the limit cycle and chaotic systems.

Engineering synchronization of chaotic oscillators using controller based coupling design

We propose a general formulation of coupling for engineering synchronization in chaotic oscillators for unidirectional as well as bidirectional mode. In the synchronization regimes, it is possible to amplify or to attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, Rössler oscillator, and a Sprott system. We physically realized the controller based coupling design in electronic circuits to verify the theory. We extended the theory to a network of coupled oscillators and provided a numerical example with four Sprott oscillators.

Bursting dynamics in a population of oscillatory and excitable Josephson junctions.

We report an emergent bursting dynamics in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducting device is considered for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. For a coupling value above a threshold, the network splits into two clusters when a reductionism approach is applied to reproduce the bursting behavior of the large network. The excitable junctions effectively induce a slow dynamics on the oscillatory units to generate parabolic bursting in a broad parameter space. We reproduce the bursting dynamics in a mixed population of dynamical nodes of the Morris-Lecar model.

Coherent libration to coherent rotational dynamics via chimeralike states and clustering in a Josephson junction array

An array of excitable Josephson junctions under a global mean-field interaction and a common periodic forcing shows the emergence of two important classes of coherent dynamics, librational and rotational motion, in the weaker and stronger coupling limits, respectively, with transitions to chimeralike states and clustered states in the intermediate coupling range. In this numerical study, we use the Kuramoto complex order parameter and introduce two measures, a libration index and a clustering index, to characterize the dynamical regimes and their transitions and locate them in a parameter plane.

Perfect synchronization in networks of phase-frustrated oscillators

Synchronizing phase frustrated Kuramoto oscillators, a challenge that has found applications from neuronal networks to the power grid, is an eluding problem, as even small phase-lags cause the oscillators to avoid synchronization. Here we show, constructively, how to strategically select the optimal frequency set, capturing the natural frequencies of all oscillators, for a given network and phase-lags, that will ensure perfect synchronization. We find that high levels of synchronization are sustained in the vicinity of the optimal set, allowing for some level of deviation in the frequencies without significant degradation of synchronization. Demonstrating our results on first and second order phase-frustrated Kuramoto dynamics, we implement them on both model and real power grid networks, showing how to achieve synchronization in a phase frustrated environment.