Environmental effects on the dynamics of coupled nonlinear systems

Abstract

Most of the real world systems are complex and their dynamics is studied by modeling them by many subunits interacting with one another. Such systems with interactions are well studied in the last few decades and are found to exhibit many emergent phenomena. In real world situations, the systems are not isolated, but are in contact with some environment. Studies of nonlinear systems in interaction with the environment are very rare. There are many cases where the different dynamical activities are regulated, monitored or triggered by a common medium or environment. The actual mechanism in such cases are not fully understood and hence this forms a topic of great relevance for a detailed study.

In this thesis, we report the study of emergent phenomena such as synchronization and amplitude death in nonlinear dynamical systems caused by coupling via a common shared environment. We first consider a simple case where, an environment interacts with two uncoupled chaotic systems. The environment has an intrinsic damped dynamics of its own, which is modulated via feedback from the systems. The environment in turn, gives feedback to both the systems. Taking standard chaotic systems such as R\"ossler and Lorenz systems as examples, we show that such an interaction can be tuned to induce a variety of synchronization phenomena such as in-phase, anti-phase, complete and antisynchronization.

We then consider a case where, two systems are coupled directly such that with sufficient strength of coupling, they can exhibit synchronous behaviour. The indirect feedback coupling through the environment is introduced in them in such a way as to induce a tendency for anti-synchronization. We show that, for sufficient strengths, these two competing effects can lead to amplitude death. By choosing a variety of dynamics such as periodic, chaotic, hyperchaotic and time-delay systems, we illustrate that this mechanism is quite general and works for different types of direct coupling such as diffusive, replacement and synaptic and different damped dynamics for environment. The specific systems considered in this study are chaotic R\"ossler and Lorenz systems, Landau-Stuart and van der Pol oscillators, hyperchaotic systems such as hyperchaotic R\"ossler and Mackey-Glass time-delay systems, non-autonomous systems such as driven van der Pol and Duffing systems.

We further extend the study to the case of a complex network of nonlinear oscillators, in interaction with a common environment. We study the onset of amplitude death in networks of R\"ossler systems and Landau-Stuart oscillators. An important result of our analysis is that there exists a universal relation between the critical value of coupling strength for amplitude death and the largest non-zero eigenvalue of the coupling matrix for different network topologies.

We develop the stability analysis for all these cases independent of system dynamics and network topology to obtain the criteria for transition to synchronization or amplitude death in each case. Extensive numerical simulations are provided to illustrate the generality of the mechanisms. The results from direct numerical simulations are found to support the results from the stability analysis.

We apply the theory developed in the general case, to the special case of neuronal systems. using Hindmarsh-Rose system as the nodal dynamics, we study in detail the occurrence of synchronized and suppressed neural activity in neurons coupled via gap-junctions and synapses.