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

Banerjee, Soumitro; Karthik, M.S.; Yuan, Guohui; Yorke, James A.

doi:10.1109/81.847870

Cited by 244 publications

(112 citation statements)

References 29 publications

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“…23,25 By contrast, we consider a situation where an attracting fixed point changes into a spirally repelling fixed point as it moves across the border.…”

Section: The Piecewise Linear Normal Formmentioning

confidence: 99%

“…Among other phenomena, Nusse and Yorke emphasized the possibility of observing peculiar bifurcations such as direct transitions from period-2 to period-3 dynamics or from regular periodic motion into chaos. Banerjee et al [23][24][25] developed the theory of border-collision bifurcations in oneand two-dimensional piecewise smooth maps, and illustrated its application in power electronic systems. It was shown that the so-called corner-collision, sliding and grazing bifurcations all belong to this class.…”

Section: Introductionmentioning

confidence: 99%

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Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Zhusubaliyev

Mosekilde

Maity

et al. 2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewiselinear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation. Application of nonlinear dynamics and chaos theory in practice, particularly in engineering, often leads to the analysis of piecewise-smooth systems. Low-dimensional models of such systems have shown that they can exhibit behavioral transitions, referred to as border-collision bifurcations, that are qualitatively different from the bifurcations we know for smooth dynamical systems. In particular, these bifurcations can produce direct transitions from periodicity to chaos or, for instance, from period-2 to period-3 dynamics. The purpose of this article is to show that torus birth bifurcations (transitions to quasiperiodicity) can also occur via border-collision bifurcations. In this case a pair of complex conjugate Floquet multipliers jump from the inside to the outside of the unit circle. We also examine the border-collision bifurcations through which the ergodic torus is transformed into a resonance torus. Torus destruction represents one of the most complicated routes to chaos, and the possible mechanisms for torus destruction in nonsmooth systems have not yet been examined in detail. A second purpose of the present article is to initiate this analysis. Finally, we illustrate our results through a practical example from power electronics and present the first experimental verification of torus birth via a border-collision bifurcation.

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“…23,25 By contrast, we consider a situation where an attracting fixed point changes into a spirally repelling fixed point as it moves across the border.…”

Section: The Piecewise Linear Normal Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Zhusubaliyev

Mosekilde

Maity

et al. 2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Figure 11͑a͒ shows the phase portrait after the perioddoubling bifurcation. The phase portrait contains the points of the unstable cycles with the periods 4,8,12,16,24,32, and 40. As mentioned above, these cycles arise through a border-collision bifurcation occurring at the same parameter value as the period doubling, L = L * .…”

Section: B Period-doubling Border-collision Routementioning

confidence: 99%

Transitions from phase-locked dynamics to chaos in a piecewise-linear map

Zhusubaliyev

Mosekilde

et al. 2008

Phys. Rev. E

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Recent work has shown that torus formation in piecewise-smooth maps can take place through a special type of border-collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle. Transitions from an ergodic to a resonant torus take place via border-collision fold bifurcations. We examine the transition to chaos through torus destruction in such maps. Considering a piecewise-linear normal-form map we show that this transition, by virtue of the interplay of border-collision bifurcations with period-doubling and homoclinic bifurcations, can involve mechanisms that differ qualitatively from those described by Afraimovich and Shilnikov.

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“…From Appendix A, we have the explicit expression of all the stable k-cycles, which the map f(X n ) can have, and from the first equation in Eqs. (22), via the reverse function…”

Section: B Stable K-cycles and Chaosmentioning

confidence: 99%

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

Gardini

Fournier-Prunaret

Chargé

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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A unified approach to attractor reconstruction Chaos: An Interdisciplinary Journal of Nonlinear Science 17, 013110 (2007); 10.1063/1.2430294 Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit Laura Gardini, 1,a) Daniè le Fournier-Prunaret, 2,b) and Pascal Chargé 2,c) In recent years, the study of chaotic and complex phenomena in electronic circuits has been widely developed due to the increasing number of applications. In these studies, associated with the use of chaotic sequences, chaos is required to be robust (not occurring only in a set of zero measure and persistent to perturbations of the system). These properties are not easy to be proved, and numerical simulations are often used. In this work, we consider a simple electronic switching circuit, proposed as chaos generator. The object of our study is to determine the ranges of the parameters at which the dynamics are chaotic, rigorously proving that chaos is robust. This is obtained showing that the model can be studied via a two-dimensional piecewise smooth map in triangular form and associated with a one-dimensional piecewise linear map. The bifurcations in the parameter space are determined analytically. These are the border collision bifurcation curves, the degenerate flip bifurcations, which only are allowed to occur to destabilize the stable cycles, and the homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in 1-piece.

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

Cited by 244 publications

References 29 publications

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Transitions from phase-locked dynamics to chaos in a piecewise-linear map

Border collision bifurcations in a two-dimensional piecewise smooth map from a simple switching circuit

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