Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Abstract

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.

Shrinking point bifurcations of resonance tongues for piecewise-smooth, continuous maps

“…In some cases (e.g., circle maps), the periodic solutions in a tongue can be characterized by their rotation number. As was first observed for a onedimensional, piecewise-linear, circle map [15], two-parameter pictures of resonance tongues for piecewise-smooth, continuous maps typically exhibit a distinctive lens-chain (or sausage) geometry [16,17,1,18,19], as in Fig. 1.…”

“…The border-collision bifurcations occurring on resonance tongue boundaries are described by piecewise-affine maps of the form (18). Th.…”

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

“…In fact, from Proposition 4, we have that the bifurcations are only those occurring in the cycles of the map in Eq. (23). We have so proved (by using Appendix A) the following result:…”

Experimental studies on emergency processes in real time Impulse Power Conversion Systems

“…При этом в них можно выделить несколько недостатков, таких как высокий уровень пульсаций выходного напряжения и низкая электромагнитная совмести-мость, а также возможность возникновения нежелательных нелинейных явлений, которые усу-губляют указанные недостатки и способствуют выходу из строя системы питания и всего обору-дования в целом [1]. Для исследования динамики ИСПЭ требуется экспериментальная система, которая позволяет автоматизировано менять основные параметры силовой части и регулятора в широких пределах, а также апробировать алгоритмы по прогнозированию, идентификации и устранения нежелательных нелинейных явлений в режиме реального времени.…”