Dynamics of homoclinic tangles in periodically perturbed second-order equations

Abstract

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 Λ μ 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 d 2 q d t 2 + ( λ − γ q 2 ) d q d t − q + q 2 = μ q 2 sin ω t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation.