Bifurcation et symetrie dans les systemes dynamiques discrets couples.

Abstract

This thesis presents the results of the theoretical study of a system that produce chaos. The system is modeled by a coupled transformation (denoted T) in two dimensions based on a sine function. In this study, we analyze first the stability ; the system bifurcations (fold and flip bifurcation when multipliers associated with the cycle of order k crosses the value +1 and −1 respectively) attractors and their basins of attraction. Afterwards, with analyticalnumerical methods, we construct the invariant varieties passing through points of type col. The transformation is not invertible (TNI) ; we also determine the critical lines (LC) of the system. The first chapter of this thesis is devoted to the recall of a number of elementary notions concerning point transformations, then to the definition of more specific notions (invariant curves, basins of attraction, and critical lines.) Reminders for conventional bifurcations (fold, transcritical or exchange of stability, pitchfork and flip) are given. In the second chapter, we conducted a joint study, theoretical and numerical, for the coupled transformation T(a,b,c) which generates very complex structures, we mainly used a numerical technique to determine the global structures of bifurcation with stability domain called "Scan", and a digital program "FORTRAN" to trace the bifurcation curves in the parametric plan (a, b) when the third parameter 0 ≤ c ≤ 1 varies. In the third chapter, we study the same transformation T(a,b,c) in the phase plan (x, y) ; we treat some problems related to either the existence of invariant curves through a fixed collar point, or to the basins of attraction. We expose the critical lines, and finallywe determine the absorbing zones. We conducted a digital program "FORTRAN" to trace basins of attraction of different present singularities, then we showed and studied qualitative changes in behavior by varying the system parameters.