Chaotic Behavior in Nonautonomous Equations without Any Time Periodicity

We investigate chaotic behavior of ordinary differential equations with a homoclinic orbit to a dissipative saddle point under a general time dependent forcing without any periodicity in time. We study Poincaré return maps in extended phase space, introducing a characteristic function that generalizes the classic Melnikov function. We then show that the dynamics of the solutions of these equations are largely determined by asymptotic behavior of this new characteristic function. We prove the existence of a spectrum of dynamical scenarios including (i) an attracting integral manifold; (ii) intersections of the stable and the unstable manifolds of the perturbed saddle point; and (iii) new dynamical structures that generalize Smale’s horseshoes for time-periodic equations. In particular, intersections of the stable and unstable manifold of the perturbed saddle are neither necessary nor sufficient for chaotic dynamics to emerge. These results are also applied to Duffing’s equation with a time dependent forcing.