Generic-type results for chaotic dynamics in equations with periodic forcing terms
Via an approach based on global variational methods, this paper intends to prove that a class of equations containing the classical periodically forced pendulum problem displays the main features of chaotic dynamics for a set of forcing terms open and dense in suitable spaces. Chaotic dynamics is defined here when the solutions satisfy the four following conditions: they depend sensitively of initial conditions, infinitely many periodic solutions with increasing period exist, ditto for an uncountable number of bounded nonperiodic solutions, and the Poincaré map of this problem has positive topological entropy. It must be noted that the two-dimensional nonautonomous equations considered here are conservative that is there is no damping term. Being a generalization of the pendulum model without damping, the restoring force is a periodic function of the angular variable, i.e., the phase space of such problems is cylindrical. This restoring force satisfies two hypotheses equivalent to those verified by the sin-function in the pendulum case. The authors' main result is implicitely contained in Birkhoff's works in the more general case of two-dimensional conservative equations with periodic terms in time. Concerning the existence of ``instability rings'', they were obtained in the first half of the last century. Such Birkhoff's instability rings are related to homoclinic and heteroclinic points generated by the associated Poincaré map. It would have been good that the authors quote this basic contribution and so compare the respective advantages of each approach.
scientific article; zbMATH DE number 1797537
scientific article