Bifurcation phenomena in non-smooth dynamical systems

R.I. Leine, D.H. van Campen
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="" style="color: black;">European Journal of Mechanics. A, Solids</a> </i> &nbsp;
The aim of the paper is to give an overview of bifurcation phenomena which are typical for non-smooth dynamical systems. A small number of well-chosen examples of various kinds of non-smooth systems will be presented, followed by a discussion of the bifurcation phenomena in hand and a brief introduction to the mathematical tools which have been developed to study these phenomena. The bifurcations of equilibria in two planar non-smooth continuous systems are analysed by using a generalised
more &raquo; ... an matrix. A mechanical example of a non-autonomous Filippov system, belonging to the class of differential inclusions, is studied and shows a number of remarkable discontinuous bifurcations of periodic solutions. A generalisation of the Floquet theory is introduced which explains bifurcation phenomena in differential inclusions. Lastly, the dynamics of the Woodpecker Toy is analysed with a one-dimensional Poincaré map method. The dynamics is greatly influenced by simultaneous impacts which cause discontinuous bifurcations. 1. Non-smooth continuous systems which are described by differential equations with a continuous but nondifferentiable right-hand side. 2. Filippov systems which are described by differential equations with a discontinuous right-hand side, but with a time-continuous state. Systems of this type can be transformed into differential inclusions with a set-valued right-hand side by using Filippov's convex method (Filippov, 1988) . 3. Systems which expose discontinuities in time of the state, such as impacting systems with velocity reversals and electrical systems with resets. This type of systems can be described by measure differential inclusions. Nonlinear dynamical systems can possess equilibria and periodic solutions as well as other special solutions of the system such as quasi-periodic solutions and chaotic attractors/repellors, which can all be either stable or unstable, and determine the long-term dynamics of the system. Here, we will consider equilibria and periodic solutions of non-smooth dynamical systems. It is often desirable to know how the equilibria and periodic solutions of a system alter when a parameter of the system is varied. The number and stability of equilibria/periodic solutions can change at a certain critical parameter value. Loosely speaking, this qualitative change in the structural behaviour of the system is called bifurcation, a word introduced by H. Poincaré. Various definitions of bifurcation exist in literature, which all agree for smooth dynamical systems, but become distinct when applied to non-smooth systems (Leine and Nijmeijer, 2004) . The most intuitive definition of bifurcation is to say that a bifurcation occurs when the number of equilibria/periodic solutions changes at a critical parameter value, being what Poincaré originally meant with the word bifurcation. In this paper, we will adopt this definition as it can be unambiguously applied to non-smooth systems. The theory of bifurcations in smooth dynamical systems is well understood, but much less is known about bifurcations in non-smooth systems although currently a lot of research is being done on this topic. Bifurcations of equilibria of non-smooth continuous systems are related to bifurcations of fixed points of piecewise smooth maps. Nusse and York (1992) study so-called 'border-collision bifurcations' of two-dimensional non-smooth discrete maps. Many publications deal with bifurcations in non-smooth systems of Filippov-type. Published bifurcation diagrams are often constructed from data obtained by brute force techniques and only show branches of stable periodic solutions (see for instance
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1016/j.euromechsol.2006.04.004</a> <a target="_blank" rel="external noopener" href="">fatcat:ddqxl47ijvaotj3d7cvtn6q6fi</a> </span>
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