Bifurcations of Tori in 3D Nonlinear Discrete-time Systems

Mondal, Ananya (2017) Bifurcations of Tori in 3D Nonlinear Discrete-time Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Ananya Mondal (12MS029)) MS_Thesis_AnanyaMondal.pdf - Submitted Version Restricted to Repository staff only Download (7MB)

Abstract

A Poincaré section on a torus in continuous time gives a closed invariant curve or simply, a loop in discrete-time. A loop can double in two different ways - (a) torus doubling resulting in two disjoint loops (b) torus doubling to form a single closed curve with two loops. There is ambiguity in the theory that explains the creation of two disjoint loops. In this study we try to resolve this issue. On the other hand, a Poincare′ section on a resonant torus gives a mode-locked loop in discrete-time. We ask the same question, in how many ways can the mode-locked loop double. It again has two possibilities- two disjoint loops or a single closed curve with two loops, known as M¨obius strip. In this study, we explain how period-doubling and Neimark-Sacker bifurcation mechanisms can lead to simultaneous doubling of nodes and saddles on the mode-locked loop which in turn leads to loop-doubling. The eigenvalue analysis helps us to distinguish between the topological difference between the two types of loop-doubling. We also report physical systems where these phenomena are observed. Apart from these, we have reported cases where either the nodes or the saddles double. Such cases give different topological structures. Therefore, the main objective of the study is to exemplify the existing theory and understand the underlying dynamics for bifurcations in tori in three-dimensional discrete-time systems.

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