Normal Lyapunov Exponents And Asymptotically Stable Attractors

In this thesis, We want to study the relation between the normally uniformly hyperbolism and the normally nonuniformly hyperbolism mainly by the tool of normal Lyapunov exponents. We discuss three different problems in the case of transformations and flows respectively.We firstly consider the synchronization in direc tionally coupled systems generated by map. Under the assumption of normally noiiu:iiformly contraction, we prove the continuousity of the synchronization mapping.Next we are concerned with the problem on the global attraction of lower-dimensional attractors:if the largest normal Lyapunov exponent is less than zero for every ergodic measure, then the attractor could be a higher-dimensional one. This generalize the result given by Afraimovich etc in 1996. Moreover, we get this proposition: the largest normal Lyapunov exponent can obtain its maximum on the set of all ergodic invariant measures(this set may not be compact).Finally we study the invariant submanifolds for flows. How can it persist under perturbations? Here we extend the conditions given by Josic in 2000 but get the same conclusion.