Beitrag in Buch INBOOK-2005-02

Bibliograph.
Daten
Avrutin, V.; Schanz, M.; Levi, P.: On multi-parametric bifurcations in piecewise-smooth dynamical systems.
In: Schöll, E. (Hrsg); Lüdge, K. (Hrsg): Book of Abstracts. Vol. 29 E.
Universität Stuttgart, Fakultät Informatik, Elektrotechnik und Informationstechnik.
XXV Dynamics Days Europe 2005, S. 25-28, englisch.
Berlin, Germany: European Physical Society, Juli 2005.
Beitrag in Buch.
CR-Klassif.G.1.10 (Numerical Analysis Applications)
Kurzfassung

Dynamical systems with a discontinuous system function represent a central topic of many scientific works published in the recent years. This research area is motivated by several practical applications, ranging from power electronic circuits (for instance DC/DC converters) to mechanical systems with impact or stick-slip phenomena, as well as hybrid and relay controlled systems. In the field of nonlinear dynamics 1D maps with a piecewise-smooth system function are well-known as return maps, obtained by the investigation of Poincaré sections of dynamical systems continuous in time. The discontinuity of these return maps is caused by the stretching, squeezing and folding phenomena, which are inherent for chaotic attractors. This discontinuity represents a border in the state space, which leads to the basic property of piecewise-smooth dynamical systems, namely their ability to undergo border-collision bifurcations.

Multi-parametric bifurcations (also known as co-dimension-n bifurcations with n>1) are bifurcations, which can be adequately described only in an n-dimensional parameter space. The characteristic property of these bifurcations is, that at these points a number m>1 of (n-1)-dimensional bifurcation hyper-surfaces (for instance, curves for n=2) intersect each other. The fundamental consequence of this property is, that a single bifurcation point of this type can dominate the dynamic behavior of the investigated system in a large area of the n-dimensional parameter space. Until now these phenomena were mostly investigated for smooth dynamical systems. Several types of these bifurcations are known, like cusps, double-Hopf, Shilnikov-Hopf, etc., which are induced by simple local bifurcations (saddle-node, Hopf). In piecewise-smooth dynamical systems, multi-parametric bifurcations can be induced by border-collision bifurcations as well. In this work, we investigate an important special case of such a bifurcation, where the aforementioned number m is infinite.

Therefore, a 1D dynamical system, discrete in time, with a single point of discontinuity is considered. This map has 3 parameters and represents some kind of normal form for maps with discontinuous system function. The part of the 3D parameter space, where this map shows periodic dynamics, is investigated in detail. It is demonstrated, that the behavior of the investigated map is determined by border collision bifurcations, whereby the areas in the parameter space leading to specific limit cycles can be obtained analytically. The main result of the presented work is a detailed description of the multi-parametric bifurcations in the investigated piecewise-linear map. The complete structure of the 3D parameter space is described, and it is shown, that the investigated map shows not only the usual one-parametric bifurcations, but also an infinite number of two-parametric bifurcations. Additionally, it is shown, that in the investigated system a three-parametric bifurcation occurs. It turns out, that the complete structure of the 3D parameter space is reflected in the structure of an infinite small vicinity of this bifurcation point. In other words: the complete structure of the 3D parameter space can be explained and described by the investigation of the single point, where this three-parametric bifurcation occurs.

Abteilung(en)Universität Stuttgart, Institut für Parallele und Verteilte Systeme, Bildverstehen
Eingabedatum30. Oktober 2005
   Publ. Institut   Publ. Informatik