Vassili GELFREICH

Institut fur MathematikFreie Universitat Berlin
Arnimallee 2-6
D-14195 Berlin
Germany
gelf@maia.ub.es
 

Examples of exponentially small transversality of separatrices

Abstract: The condition, which ensure the exponential smallness of the separatrices splitting, are rather well known. On the other hand establishing the transversality of the separatrices for such systems is an extremely difficult problem. In the talk some examples are described. Each system has a transversal homoclinic trajectory and the corresponding splitting is exponentially small with respect to the perturbation parameter. This includes the famous standard map, periodic and quasiperiodic perturbations of the pendulum and some other examples. The special attention is given to comparing of the actual value of the splitting with the Poiancare-Arnold- Melnikov theory.