Date of Award

29-7-2024

Document Type

Thesis

School

School of Electrical & Electroncis Engineering

Programme

Ph.D.-Doctoral of Philosophy

First Advisor

Dr.V.K.Chandrasekar

Keywords

Symmetry Breaking Coupling, Coupled Systems, Aging Transition

Abstract

The knowledge of emergent dynamics and the mechanisms by which self-organizing complex systems operate is a perennially difficult problem that can be effectively resolved by employing an ensemble of coupled dynamical units as a framework. Several intriguing collective dynamical behaviors, such as synchronization, chimera, clustering, oscillation quenching, etc., have been reported using an ensemble of coupled oscillators. Failure or degradation in the performance of even one of the local oscillators in a network of oscillators can lead to cascading failures resulting in the collapse of the entire network such as power blackouts. Similar dynamics can also be observed among neuronal networks despite billions of neurons being born and dying daily.

For instance, neuronal pathological disorders such as Alzheimer’s disease are mainly due to the degradation/failure of neurons in assemblies. The report of Daido and Nakanishi has led to a flurry of research activities on the phenomenon of the aging transition. In the beginning stage, disorder-induced aging transition was reported in locally coupled oscillators, and the study was extended to a various coupling configurations. In particular time-delay effects on the aging transition in a large population coupled with nonlinear oscillators have also been reported. The aging transitions are analyzed so far only by adapting symmetry preserving global or locally coupled oscillators.

In this thesis, we investigate the macroscopic dynamics of networked oscillators, focusing on how the deterioration or failure of microscopic constituents affects the overall activity and stability of the heterogeneous network coupled via symmetry-breaking interaction. We begin by analyzing a globally coupled network of heterogeneous oscillators, deriving the evolution equation for two macroscopic order parameters. Then, we explore the dynamical robustness of the network by introducing limiting factors such as diffusion and self-feedback.

We deduce the evolution equation of two macroscopic order parameters from a globally coupled network of heterogeneous oscillators following the self-consistent field approach under the strong coupling limit. Further, we explore that the symmetry-breaking coupling facilitates the onset of the homogeneous steady state among the population at the critical proportion of the inactive oscillators despite large number of active oscillators. Interestingly, a chimera-like death state is observed in the study related to the aging transition for the first time in the literature.

The competing effect of heterogeneity and symmetry-breaking coupling on the emerging dynamics in a system of N globally coupled Stuart-Landau oscillators is investigated. Increasing the heterogeneity, Hopf bifurcation parameter’s standard deviation favors the macroscopic oscillatory state for low values of the symmetry-breaking coupling and the inhomogeneous steady state for larger coupling values. There is also a transition, tipping, to a homogeneous steady state (aging state) from the macroscopic oscillatory state.

Analytical stability (critical) curves of these bifurcations, deduced from the mean-field variables. From the result, we conclude that heterogeneity and symmetry-breaking coupling plays a crucial role in the aging phenomenon. Moreover, we find that the symmetry-breaking coupling can facilitate the onset of the aging transition, whereas the large heterogeneity factor facilitates the stable macroscopic oscillatory state by destabilizing the aging transition state. The aforementioned outcomes have given rise to numerous phenomena in neural networks and power grids.

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