Method of Bifurcation Analysis of Hybrid Nonlinear Systems

Abstract

I. INTRODUCTION Systems with continuous and discrete dimensions, i.e. hybrid systems, exist in many domains such as mechanical or electronic systems. The solution function is usually continuous but presents points of non-derivability where discrete changes occur. Some of the most widely studied applications are related to electrical engineering, such as power converters investigated as piecewise smooth systems by di Bernardo [1], Tse [2] or Banerjee [3][4]; or some PLL models as introduced By Acco [5], refering to this type of model as " hybrid sequential. " Little work has been done in the analysis of nonlinear hybrid models, apart from Kawakami, Ueta and Kousaka [7][8]. As for hybrid linear models, they can be analysed using rigorous analytical methods as did Kabe [6]. But the nonlinear property of many systems reserve this option for approximated models only. Instead, we prefer using numerical methods. We review the analysis method based on a Poincaré map and introduce some modifications and improvements. We consider the analysis results of an extended version of the Alpazur oscillator, using a numerical process to approximate some derivatives (adapted to the automated CAD tools we are working on). But the most compelling part is to be found in our example results, revealing an unusual bifurcation structure: the interactions between the equilibrium point at some state and the corresponding switching condition make appear a fractal bifurcation structure, with an infinite number of bifurcation curves focusing towards a limit set. As this is due to the hybrid properties of the system, we may find such structure in many other hybrid systems.

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Cite this paper

@inproceedings{Brandon2008MethodOB, title={Method of Bifurcation Analysis of Hybrid Nonlinear Systems}, author={Quentin Brandon and Tetsushi Ueta and Daniele Fournier-Prunaret}, year={2008} }