Journal ArticleDOI
Border collision bifurcations in two-dimensional piecewise smooth maps
Soumitro Banerjee,Celso Grebogi +1 more
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In this article, a theoretical framework for the classification of border collision bifurcations is presented, which can help in explaining bifurbation in all systems, which are represented by two-dimensional piecewise smooth maps.Abstract:
Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps that cannot be classified among the generic cases like saddle-node, pitchfork, or Hopf bifurcations occurring in smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form - the piecewise linear approximation in the neighborhood of the border. We show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form, and these are classified under six cases. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. This theoretical framework will help in explaining bifurcations in all systems, which can be represented by two-dimensional piecewise smooth maps.read more
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Complex Behavior of Switching Power Converters
TL;DR: In this article, the authors focus on the typical bifurcation scenarios and nonlinear behavior observed in switching power converters, expounding on their most relevant aspects from a practical viewpoint.
Journal ArticleDOI
Complex behavior in switching power converters
Chi K. Tse,M. Di Bernardo +1 more
TL;DR: This paper provides an overview of the chaotic dynamics and bifurcation scenarios observed in power converter circuits, emphasizing the salient features of the circuit operation and the modeling strategies.
Journal ArticleDOI
Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits
TL;DR: In this paper, the authors present a systematic analysis of the bifurcation behavior of power electronic DC-DC converters through a normal form: the piecewise linear approximation in the neighborhood of the border.
Journal ArticleDOI
Bifurcations in Nonsmooth Dynamical Systems
Mario di Bernardo,Chris Budd,Alan R Champneys,Piotr Kowalczyk,Arne Nordmark,Gerard Olivar Tost,Petri T. Piiroinen +6 more
TL;DR: A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems, to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillator, DC-DC converters, and problems in control theory.
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Dynamics and bifurcations of nonsmooth systems: A survey
TL;DR: In this article, the main focus is on piecewise smooth systems, which have recently attracted a lot of attention, but also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities.
References
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Chaos: An Introduction to Dynamical Systems
TL;DR: One-dimensional maps, two-dimensional map, fractals, and chaotic attraction attractors have been studied in this article for state reconstruction from data, including the state of Washington.
Journal ArticleDOI
Border-collision bifurcations including “period two to period three” for piecewise smooth systems
TL;DR: In this paper, the authors examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter μ, and they show that stable periodic orbits occur frequently in such situations, but never or almost never occur in smooth systems.
Journal ArticleDOI
Modeling of chaotic DC-DC converters by iterated nonlinear mappings
TL;DR: In this paper, the authors describe DC-DC converters by iterated mappings, a nonlinear discrete modeling technique that is applied to the example of a PWM-controlled buck converter.
Journal ArticleDOI
Instability, subharmonics and chaos in power electronic systems
J.H.B. Deane,D.C. Hamill +1 more
TL;DR: In this article, the concept of chaos is applied to a variety of nonlinear power electronic circuits and the phenomena of subharmonics, quasi-periodicity, and chaos are predicted and observed.
Journal ArticleDOI
Grazing bifurcations in impact oscillators
TL;DR: This paper analyzes the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place.
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