Border-Collision Bifurcations in Piecewise-Smooth Dynamical Systems
Zh. Zhusubaliyev, E. Soukhoterin, C. Knudsen, and E. Mosekilde
Many systems of practical interest display non-smooth variations in the functional relationships. In mechanical engineering, for instance, this is the case for systems that involve friction, collisions or finite clearances. Besides friction, the motion of a railway wheelset involves abrupt changes in the horizontal forces when there is flange contact between the wheels and the rails. Friction forces by themselves typically display discontinuous variations at vanishing speeds.
Discontinuities are also observed in relay control systems or in systems with mechanical valves. Each time the relay switches or a valve closes or opens, the system starts on a new trajectory. The controlled system may be linear by itself. However, using appropriate sewing conditions, the trajectory has to be connected across the switching events. Human decision making behavior may lead to a similar type of abrupt changes in the working conditions of a system, and the same is truth for the intervention thresholds that play a significant role in attempts to stabilize various economic markets.
Besides the usual local (saddle-node, period-doubling and Hopf) bifurcations that we know from smooth dynamical systems, piecewise-smooth dynamical systems display a qualitatively different type of bifurcations, so-called border-collision bifurcations, that are related to the crossing of the trajectory into new regions of operation for the system.
Border-collision bifurcations are characterized by abrupt jumps in the eigenvalues of the various periodic states. Hence, one can observe direct transitions to chaos as well as period-tripling, period-quadrupling, etc. bifurcations. Border-collision bifurcations are also characterized by a nearly linear (by contrast to the usual parabolic) growth of the amplitude of the emerging mode after a bifurcation.
At the same time the classical local bifurcations change form. For a piecewise-linear system, for instance, the first period-doubling transtion becomes abrupt, and the system typically shows a direct transition to four-band chaos in the following bifurcation. Moreover, many of the periodic windows in the chaotic regions are missing.
The aim of the present project is two-fold: We want to examine the characteristic features of border-collision bifurcations, and we want to illustrate the importance of such phenomena in modern electrotechnical control systems.
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The figure on this page illustrates the synchronization regions in a multi-dimensional piecewise-smooth map modelling a power electronic DC/DC converter.