Facta Univ. Ser.: Elec. Energ., vol. 24, No. 1, April 2011, pp. 107-119

Dynamics of Three Dimensional Maps

Asma Djerrai and Ilhem Djellit

Abstract: Smooth 3D maps have been a focus of study in a wide range of research fields. Their Properties are investigated qualitatively and numerically. These maps show qualitatively interesting types of bifurcations than those exhibited by generic smooth planar maps. We present a theoretical framework for analyzing three-dimensional smooth coupling maps by finding the stability criteria for periodic orbits and characterizing the system behaviors with the tools of nonlinear dynamics relative to bifurcation in the parameter plane, invariant manifolds, critical manifolds,chaotic attractors. We also show by numerical simulation bifurcations that can occur in such maps. By an analytical and numerical exploration we give some properties and characteristics, since this class of three-dimensional dynamics is associated with the properties of one-dimensional maps. There is an interesting passage from the one-dimensional endomorphisms to the three-dimensional endomorphisms.

Keywords: Three-dimensional maps, bifurcations, invariant closed curve.