Dynamics and bifurcations in nonlinear systems with hysteresis

Abstract

The main objective of this thesis is to determine periodic solutions and possible bifurcations in a family of dynamical systems with hysteresis. In par- ticular, we analyze a concrete family of slow-fast piecewise linear systems in three dimensions, which after introducing a hysteretic function, is embedded in two dimensions. In Chapter 1, we introduce the tridimensional piecewise linear systems to be studied. Then, we reduce their dimension thanks to a relaxation hy- pothesis. Moreover, we show a normalized canonical form which allows us to decrease the number of parameters. In Chapter 2, we analyze the case where the hysteretic system has no isolated equilibria, that is, when one of the eigenvalues of the planar system is zero. We detect some relevant bifurcations as a saddle-node bifurcation of periodic orbits and a bifurcation from in nity. In Chapters 3 and 4, we study the non-zero real eigenvalues cases, giving rise to equilibria of node, saddle or improper node type. Bifurcations as saddle-node of periodic orbits, homoclinic and heteroclinic bifurcations are analytically described. We detect also, some regions on the parameter plane where the existence of four periodic orbits is possible. In Chapter 5 we focus on the complex eigenvalues case, that is, when the equilibria are centre or focus type. For the centre case, the dynamics is completely determined and we can detect similar bifurcations as the obtained in previous chapters. The focus case is tackled in a diferent way. In this last case, our aim will be to obtain conditions for which the parameters lead to an instantaneous transition to chaotic behaviour. Finally, after a conclusions chapter, we include some non-generic cases in appendices A and B.