DYNAMICAL COMPLEXITY NEAR NON-CONTROLLABLE 3D PIECEWISE LINEAR LUR’E SYSTEMS Victoriano Carmona, Emilio Freire, Enrique Ponce, Francisco Torres Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos, 41092-Sevilla, Spain, e-mail: {vcarmona, efrem, eponcem, ftorres}@us.es Abstract: A very useful approach to understand the complex behavior of a system is to consider its degenerate situations. Accordingly, the structure of periodic orbits for some degenerate symmetric piecewise linear three–dimensional systems with three zones, whose linear parts share a pair of imaginary eigenvalues is analyzed. Under this hypothesis, the system is noncontrollable and its study can be reduced to the analysis of a periodic one–dimensional equation that can have up to five periodic orbits. Copyright c  2006 IFAC. Keywords: Piecewise linear systems, controllability, periodic orbits, complexity. 1. INTRODUCTION Bifurcation theory is a branch of the qualitative theory of dynamical systems which focuses its study at the critical values of parameters where there is a change in the qualitative behavior of the systems, see for instance (Guckenheimer and Holmes, 1985). Such critical values where there is a lack of parametric robustness represent bor- der situations whose analysis can provide a deep insight into the behavior catalog of the system. Therefore, by introducing artificial parameters if needed, and looking for their possible bifurcation values, one can discover and clarify the dynamical behavior of the system, even under nominal values of parameters and non-critical situations. Roughly speaking, it can be said that the different possi- ble standard behaviors of nonlinear systems are hidden in their bifurcation points. 1 Authors are partially supported by spanish Ministerio de Ciencia y Tecnolog´ ıa under Grant BFM2003-00336 and MTM2004-04066. In this paper, the usefulness of a bifurcation ap- proach in the analysis of the dynamical complex- ity of symmetrical 3D piecewise linear control systems is investigated. Continuous single-input single-output control systems with three linear regions will be mainly considered, for this seemed simple case is very frequent in real applications and it is enough to deal with very complex dy- namics, see (Madan, 1993). A starting point to understand the complex be- havior of a system consists in looking for degener- ate situations and perturb them. Here, the critical situation to be considered is related to the lack of controllability: it will be shown that a rich structure of periodic orbits appears associated to the corresponding bifurcation point. In its unfold- ing, a great variety of dynamical behaviors, most of them chaotic, can be found, but its complete analysis goes beyond the scope of this paper. Let us start by considering the following system in R 3 ˙ x(t)= Ax(t)+ bu(t) (1)