Investigating Piecewise Smooth Hybrid Systems

Manik, Debsankha (2013) Investigating Piecewise Smooth Hybrid Systems. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS Dissertation of Debsankha Manik (08MS02)) Debsankha_Manik-08MS02-MSThesis.pdf - Submitted Version Restricted to Repository staff only Download (730kB)

Abstract

Piecewise smooth dynamical systems are of particular interest because of their ubiquity and the plethora of novel behaviours they exhibit. There are very elegant existing frameworks for analyzing different kinds of border collision bifurcations occurring in those systems. These include the classic work on border collision of equilibria by Feigin, analysis of grazing bifurcations using the ZDM (Zero-time Discontinuity Mapping) formalism developed by Nordmark et al.and the impact map approach developed by Bernardo et al. Using the ZDM formalism, Banerjee and Kundu analyzed the soft impacting oscillator system and predicted the onset of chaos immediately following the grazing of a limit cycle except when the ratio of the damped natural frequency of the system and the forcing function frequency are in integral or half integral ratios. Experiments have roughly borne out this prediction, except for the fact that chaos has been observed to vanish in a small neighbourhood of those values predicted analytically. Here we have used the impact map formalism to derive the necessary conditions for chaos to vanish and matched the results with the experimental results. It has been reported previously that the lifetimes of transients in systems exhibiting grazing bifurcations follow a power law pattern as the parameter value approaches the grazing value. We have observed the same characteristics in a simple harmonic oscillator with hard impacts. We found that the transient lifetimes were orders of magnitude larger than what they should have been if the impacts did not occur. We have attempted to explain this behaviour by defining a quantity which represents how likely is a system to undergo impacts for a set of parameter values.

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