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, Λ_Μ contains nothing else but a horseshoe of infinitely many branches; (ii) for infinitely many disjoint open sets of Μ, Λ_Μ contains nothing else but one sink and one horseshoe of infinitely many branches; and (iii) there are positive measure sets of Μ so that Λ_Μ admits strange attractors with Sinai-Ruelle-Bowen measure. We also use the equation. d²q/dt²+(Λ-γq²)dq/dt-q+q²=Μq²sinωt to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation.