Dynamics in Networks of Delay Coupled Nonlinear Time-delay Systems

Mitra, Chiranjit (2014) Dynamics in Networks of Delay Coupled Nonlinear Time-delay Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

We consider a network of delay dynamical systems connected in a ring via unidirectional positive feedback with the delay in coupling being modulated in time. For the speci�c case of Mackey-Glass systems on the ring topology, we demonstrate the occurrence of the emergent phenomena of amplitude death, synchronization and phase- ip bifurcation in our model as the relevant parameters are tuned. Using linear stability analysis and master stability function approach we predict the region of amplitude death and synchronized states respectively in the parameter space and study the nature of transitions between the di�erent states. For a large number of systems in the same dynamical con�guration, we observe splay states, mixed splay states and phase locked clusters. For certain parameter values, we capture multi-stability between the synchronized states in our system. We implement the idea of a real digital communication system in a hybrid version of our model, where the signal from one node is digitized (sampled and quantized) before another node receives it (with no coupling delay). In this scenario, we recover the emergent phenomena of phase- ip which has previously been observed with only variation in coupling delay. Thus, we identify the correspondence between the roles played by digitization of coupling and delay in coupling. The thesis is organized as follows: In chapter 1, we introduce the problem under consideration, its scope and brie y describe the important results. In chapter 2, we brie y de�ne and introduce the various forms of time-delay systems and delay di�erential equations used to mathematically de�ne time-delay systems as well as introduce the Mackey-Glass system which has been used throughout the course of the present work. In chapter 3, we describe the basic model that has been primarily explored throughout this work and report the emergent dynamics exhibited by the model. In chapter 4, we study the nature of transitions between the emergent dynamics reported in the previous chapter. In chapter 5, we report the emergent dynamics observed for a large number of systems on the network. In chapter 6, we report the emergent dynamics observed with the digitization of coupling between the nodes of the network. In chapter 7, we describe our plan to construct an electronic circuit to serve as an experimental equivalent of our model and report the progress made in that direction. Finally, we summarize the relevant observations in chapter 8.

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