Dynamical Properties Of A Class Of Piecewise Continuous Systems

This dissertation gives necessary and sufficient conditions for 2-torus parabolic maps to be invertible,which improves P.Ashwin et al's result,and discusses the isomorphism between invertible torus parabolic maps.Periodicity of some 2-torus parabolic maps is investigated,where periodic point sets in horocyclic case are shown to be dense in the torus topology,and some semi-rational cases are shown to possess periodic points of all periods;for the integral parabolic maps on the torus,the topological entropy is zero.Furthermore,planar piecewise parabolic maps are introduced and discussed,where 2-torus parabolic maps can be viewed as simple planar piecewise parabolic maps with general plane topology.General properties such as symbolic entropy and complexity of planar piecewise parabolic maps are investigated,and their difference under the plane topology and the torus topology for the same toms parabolic maps is also discussed. The general conditions for the existence of global attractors for a class of planar piecewise parabolic maps are presented.Finally,the concept of the Conley index for piecewise continuous maps is defined, and an example for calculating the Conley index of a 1-d piecewise isometry is given.