Heteroclinic tangles in time-periodic equations

In this paper we prove that, when a heteroclinic loop is periodically perturbed, three types of heteroclinic tangles are created and they compete in the space of μ where μ is a parameter representing the magnitude of the perturbations. The three types are (a) transient heteroclinic tangles containing no Gibbs measures; (b) heteroclinic tangles dominated by sinks representing stable dynamical behavior; and (c) heteroclinic tangles with strange attractors admitting SRB measures representing chaos. We also prove that, as μ→0, the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections. The theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical Birkhoff–Melnikov–Smale method.