Global Bifurcations in Three Dimensional Piecewise Smooth Maps

Patra, Mahashweta (2018) Global Bifurcations in Three Dimensional Piecewise Smooth Maps. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

This thesis presents a variety of new results regarding global bifurcations in three dimensional piecewise smooth discrete time dynamical systems. While there are practical dynamical systems which can be modelled by three dimensional piecewise smooth maps, a very little investigation has been carried out on the dynamics and bifurcations of such maps. Most of the past studies on nonsmooth maps have considered maps which are invertible. However, it has also been found that some switching dynamical systems give rise to nonsmooth noninvertible maps. In three dimensional systems, both 1-dimensional and 2-dimensional manifold can exist. We report a mechanism of drawing 1-D and 2-D manifolds in three dimensional piecewise smooth maps both in invertible and noninvertible scenarios. In this thesis, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of the creation of tori with multiple loops may result from border collision bifurcations. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in piecewise smooth systems, but fail in some cases-especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by the change in eigenvalues of the unstable fixed points. A chaotic attractor is called robust if there is no periodic window or any coexisting attractor in some open subset of the parameter space. Such a chaotic attractor cannot be destroyed by a small change in parameter values since a small change in the parameter value can only make small changes in the Lyapunov exponents. Earlier investigations have calculated the existence and the stability conditions of robust chaos in 1D and 2D piecewise linear maps. In this work, we demonstrate the occurrence of robust chaos in 3D piecewise linear maps and derive the conditions of its occurrence by analyzing the interplay between the stable and unstable manifolds. Multiple attractor bifurcations occurring in piecewise smooth maps lead to simultaneous creation of multiple stable orbits. A peculiar feature of such a bifurcation is that, for a parameter value slightly above the bifurcation value, the coexisting attractors are arbitrarily close to each other, so that ambient noise in the system may push the system intermittently from one attractor to another. This may be damaging for practical systems as there is a fundamental uncertainty regarding which orbit the system will follow after a bifurcation. Such bifurcations are known to occur in piecewise smooth maps, which model many practical and engineering systems. So far the occurrence of such bifurcations has been investigated in the context of 2D piecewise linear maps. In this thesis, we investigate multiple attractor bifurcations in a three dimensional piecewise linear normal form map. We show the occurrence of different types of multiple attractor bifurcations in the system, like the simultaneous creation of a period-2 orbit, a period-3 orbit, and an unstable chaotic orbit; a mode-locked torus, an ergodic torus and periodic orbits; a one-loop torus and a two-loop torus; a one-loop mode-locked torus and a two-loop mode-locked torus; a one-piece chaotic orbit and a 3-piece chaotic orbit, etc. As orbits lie on unstable manifolds of fixed points, the structure of unstable manifold plays an important role to understand coexistence of attractors. We show that interplay between 1D and 2D stable and unstable manifolds plays an important role in global bifurcations that can give rise to multiple coexisting attractors. We show various ways of the occurrence of hyperchaotic orbits in 3D piecewise linear normal form maps. We show that hyperchaotic orbits can be born from a periodic orbit or a quasiperiodic orbit in various ways like: (a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a threepiece hyperchaotic orbit. We further show various attractors coexisting with hyperchaotic orbit. Moreover, we numerically calculate the existence region of hyperchaotic orbit in the parameter space. In this thesis, we also consider synchronization of nonidentical periodic oscillators for different coupling schemes and study the nature of the synchronized frequency in smooth systems. Based on numerical and experimental observations we show that for directly coupled oscillators, the synchronized frequency lies between the individual frequencies and its value does not depend on the coupling constant, while for indirectly coupled oscillators the synchronized frequency lies out of the range and depends on the strength of coupling. We explain the different frequency behaviors of directly and indirectly coupled systems by analytically deriving the expressions of synchronized frequency under certain simplifying assumptions.

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