Bifurcations in two-dimensional piecewise smooth maps: Theory and applications in switching circuits

S Banerjee, P Ranjan, C Grebogi

Research output: Contribution to journalArticle

204 Citations (Scopus)

Abstract

Recent investigations on the bifurcation behavior of power electronic dc-dc converters hare revealed that most of the observed bifurcations do not belong to generic classes such as saddle-node, period doubling, or Hopf bifurcations. Since these systems yield piecewise smooth maps under stroboscopic sampling, a new class of bifurcations occur in such systems when a fixed point crosses the border between the smooth regions in the state space. In this paper we present a systematic analysis of such bifurcations through a normal form: the piecewise linear approximation in the neighborhood of the border. We show that there can be many qualitatively different types of border collision bifurcations, depending on the parameters of the normal form, We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. We then use this theoretical framework to explain the bifurcation behavior of the current programmed boost converter.

Original languageEnglish
Pages (from-to)633-643
Number of pages11
JournalIEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications
Volume47
Issue number5
Publication statusPublished - May 2000

Keywords

  • bifurcation theory
  • chaos
  • nonlinear dynamics
  • power electronics
  • border-collision bifurcations
  • DC-DC converters
  • buck converter
  • boost converters
  • behavior
  • systems

Cite this

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title = "Bifurcations in two-dimensional piecewise smooth maps: Theory and applications in switching circuits",
abstract = "Recent investigations on the bifurcation behavior of power electronic dc-dc converters hare revealed that most of the observed bifurcations do not belong to generic classes such as saddle-node, period doubling, or Hopf bifurcations. Since these systems yield piecewise smooth maps under stroboscopic sampling, a new class of bifurcations occur in such systems when a fixed point crosses the border between the smooth regions in the state space. In this paper we present a systematic analysis of such bifurcations through a normal form: the piecewise linear approximation in the neighborhood of the border. We show that there can be many qualitatively different types of border collision bifurcations, depending on the parameters of the normal form, We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. We then use this theoretical framework to explain the bifurcation behavior of the current programmed boost converter.",
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T1 - Bifurcations in two-dimensional piecewise smooth maps

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AU - Banerjee, S

AU - Ranjan, P

AU - Grebogi, C

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Y1 - 2000/5

N2 - Recent investigations on the bifurcation behavior of power electronic dc-dc converters hare revealed that most of the observed bifurcations do not belong to generic classes such as saddle-node, period doubling, or Hopf bifurcations. Since these systems yield piecewise smooth maps under stroboscopic sampling, a new class of bifurcations occur in such systems when a fixed point crosses the border between the smooth regions in the state space. In this paper we present a systematic analysis of such bifurcations through a normal form: the piecewise linear approximation in the neighborhood of the border. We show that there can be many qualitatively different types of border collision bifurcations, depending on the parameters of the normal form, We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. We then use this theoretical framework to explain the bifurcation behavior of the current programmed boost converter.

AB - Recent investigations on the bifurcation behavior of power electronic dc-dc converters hare revealed that most of the observed bifurcations do not belong to generic classes such as saddle-node, period doubling, or Hopf bifurcations. Since these systems yield piecewise smooth maps under stroboscopic sampling, a new class of bifurcations occur in such systems when a fixed point crosses the border between the smooth regions in the state space. In this paper we present a systematic analysis of such bifurcations through a normal form: the piecewise linear approximation in the neighborhood of the border. We show that there can be many qualitatively different types of border collision bifurcations, depending on the parameters of the normal form, We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. We then use this theoretical framework to explain the bifurcation behavior of the current programmed boost converter.

KW - bifurcation theory

KW - chaos

KW - nonlinear dynamics

KW - power electronics

KW - border-collision bifurcations

KW - DC-DC converters

KW - buck converter

KW - boost converters

KW - behavior

KW - systems

M3 - Article

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SP - 633

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JO - IEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications

JF - IEEE Transactions on Circuits and Systems Part I: Fundamental Theory and Applications

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