In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system—the static VAR compensator used in electrical power systems—and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.
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September 2010
Research Article|
July 13 2010
Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps
Biswambhar Rakshit;
Biswambhar Rakshit
1Department of Mathematics and Centre for Theoretical Studies,
Indian Institute of Technology
, Kharagpur 721302, India
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Manjul Apratim;
Manjul Apratim
2Department of Physics and Astronomy,
The State University of New Jersey
, Piscataway, New Jersey 08854-8019 USA
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Soumitro Banerjee
Soumitro Banerjee
3
Indian Institute of Science Education and Research
, Nadia, West Bengal, Mohanpur 741252, India
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Chaos 20, 033101 (2010)
Article history
Received:
February 10 2009
Accepted:
April 13 2010
Citation
Biswambhar Rakshit, Manjul Apratim, Soumitro Banerjee; Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps. Chaos 1 September 2010; 20 (3): 033101. https://doi.org/10.1063/1.3422475
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