Dynamics of homoclinic tangles in periodically perturbed second-order equations

We obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second-order ordinary differential equations with dissipation. Let μ be the size of perturbation and Λμ be the entire homoclinic tangle. We prove in particular that (i) for infinitely many disjoint open sets of μ, Λμ contains nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many disjoint open sets of μ, Λμ contains nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure sets of μ so that Λμ admits strange attractors with Sinai–Ruelle–Bowen measure. We also use the equation to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation.