Observations of Nonsmooth Bifurcation Phenomena in Switching Electronic Circuits

Seth, Soumyajit (2020) Observations of Nonsmooth Bifurcation Phenomena in Switching Electronic Circuits. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Soumyajit Seth (14RS057)) 14RS057.pdf - Submitted Version Restricted to Repository staff only Download (3MB)

Abstract

This thesis presents experimental observations of different types of bifurcation phenomena in switching electronic circuits. Those bifurcations have so far been studied from a theoretical perspective. However, experimental observations of most of these bifurcation phenomena have not yet been reported. Most of the past studies on non-smooth bifurcations in piecewise smooth systems have been carried out analytically and/or numerically. The main tool, almost always, was the normal form: a piecewise linear approximation of the piecewise smooth map in the neighbourhood of the border. Here, we have constructed electronic switching circuits whose discrete-time representations would give rise to a piecewise smooth maps. These can also be approximated by the piecewise linear maps in the neighborhood of the border. We then obtain experimental evidence of different bifurcation phenomena by appropriately choosing the parameters of the electronic systems. In earlier works, occurrence of period incrementing cascades have been shown in case of 1D piecewise smooth maps. In this thesis, we have shown the experimental observation of the period incrementing cascade in an inductor-less chaos generator circuit. Multiple attractor bifurcations occurring in piecewise smooth dynamical systems may lead to potentially damaging situations. A striking feature of these bifurcations is that, in the presence of arbitrarily small noise, it may lead to fundamentally unpredictable behavior as a system parameter is varied through its bifurcation value, as the orbit may follow any of the multiple attractors born at the bifurcation point. In earlier works, the conditions of occurrence of multiple attractor bifurcation were reported in the context of a piecewise linear 2D map. In this thesis, we report the Ąrst experimental observation of this phenomenon in an electronic switching system. We have shown the simultaneous creation of a period-3 and the chaotic attractors just above the bifurcation value of this electronic switching system. Robust chaos occurring in piecewise smooth dynamical systems is very important in practical applications. It is deĄned by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. A small change in parameter values cannot destroy such a chaotic attractor since a small change in the parameter value can only make small changes in the Lyapunov exponents. In earlier works, the occurrence of robust chaos was reported in the context of piecewise linear 1D, and 2D maps and the regions of occurrence have been investigated in 1D and 2D switching circuits. In this thesis, we report the Ąrst experimental observation of this phenomenon in a 3D electronic switching system and obtained the region of parameter space by constructing a discrete map. Mechanical impacting systems exhibit a large array of interesting dynamical behaviors, including a large amplitude chaotic oscillation close to the grazing condition, called Śnarrow band chaos.Š This phenomenon has been explained by means of square root singularity and the occurrence of dangerous border collision bifurcation. However, experimental investigation in the area is constrained by the fact that the parameters of such a mechanical system cannot be varied easily. Here we propose an electronic circuit that can act as an analog of an impacting mechanical system. We show that the phenomena earlier reported through numerical simulation (like narrow band chaos, Ąnger-shaped attractor, etc.) occur in this system also. We have experimentally obtained the evolution of the chaotic attractor at grazing as the stiffness ratio is variedŮwhich is not possible in mechanical experiments. We experimentally conĄrm the theoretical prediction that the occurrence of narrow band chaos can be avoided for some parameter settings.

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