Sliding Bifurcations: A Novel Mechanism for the Sudden Onset of Chaos in Dry Friction Oscillators

Bernardo, Mario di; Kowalczyk, Piotr; Nordmark, Arne

doi:10.1142/s021812740300834x

Cited by 138 publications

(92 citation statements)

References 31 publications

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“…It was shown that the so-called corner-collision, sliding and grazing bifurcations all belong to this class. 9,[26][27][28][29][30][31][32] To investigate the various types of bifurcation, a normal form map may be constructed ͑see, e.g., Refs. 26-29͒.…”

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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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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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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show abstract

“…One of the intriguing features of hybrid systems is the possibility of a sudden onset of chaotic dynamics. Such scenarios have been observed, for instance, in the context of modelling DC/DC power converters Banerjee et al (1998); Banerjee and Verghese (2001), and dry-friction oscillators di Bernardo et al (2003). The onset of chaotic dynamics in these systems was shown to have been triggered by a nontrivial interaction between a system Ω−limit set and the so-called switching surface.…”

Section: Introductionmentioning

confidence: 91%

Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling

Kowalczyk

Glendinning

2011

IFAC Proceedings Volumes

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Abstract:We investigate a class of linear relay feedback systems with bang-bang control and with the control input applied at discrete time instances. Using a third order system as a representative example we show that stable oscillations with so-called sliding motion, with sliding present in continuous time system, loose the sliding segment of evolution, but do not loose their stability if the open loop system is stable. We then carry on our investigations and consider a situation when stable self-sustained oscillations are generated with the unstable open loop system. In the latter case a transition from a stable limit cycle to micro-chaotic oscillations occurs. The presence of micro-chaotic oscillations is shown by considering a linearised map that maps a small neighbourhood of initial conditions back to itself. Using this map the presence of the positive Lyapunov exponent is shown. The largest Lyapunov exponent is then calculated numerically for an open set of sampling times, and it is shown that it is positive. The boundedness of the attractor is ensured for sufficiently small sampling times; with the sampling time tending to zero these switchings become faster and they turn into sliding motion. It is the presence of the underlying sliding evolution that ensures the boundedness of the chaotic attractor. Our finding implies that what may be considered as noise in systems with digital control should actually be termed as micro-chaotic behaviour. This information may be helpful in designing digital control systems where any element contributing to what appears as noise should be suppressed.

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“…Early interest in discontinuity-induced bifurcations in such systems has naturally focused on regions of a switching manifold that are attractive, and where sliding occurs. The result is that periodic orbits can acquire segments of sliding by so-called sliding bifurcations (di Bernardo et al, 2003(di Bernardo et al, , 2008). The scenario of sliding in a region where solutions are repelled from the switching manifold (sometimes called escaping), appears at first to be of less interest dynamically, because one expects solutions to evolve away from the discontinuity.…”

Section: Introductionmentioning

confidence: 99%

Sliding bifurcations and non-determinism in systems with switching

Jeffrey¹

2011

IFAC Proceedings Volumes

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Dynamical events associated with loss of determinacy in piecewise-smooth dynamical systems are reviewed. The causes of non-determinacy are discussed, and the conditions for their occurence in general systems are derived. These events can be characterized in terms of recently classified catastrophic sliding bifurcations and a non-deterministic form of chaos, which are shown to provide generic scenarios for non-determinacy to affect stable dynamical behaviour. The particular conditions for these are derived in a canonical (Lur'e) model of switched feedback control, and explicit examples are given in two and three dimensions.

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Sliding Bifurcations: A Novel Mechanism for the Sudden Onset of Chaos in Dry Friction Oscillators

Cited by 138 publications

References 31 publications

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling

Sliding bifurcations and non-determinism in systems with switching

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