On the Detection and Dynamical Consequences of Orbits Homoclinic to Hyperbolic Periodic Orbits and Normally Hyperbolic Invariant Tori in a Class of Ordinary Differential Equations

Abstract

We develop a global perturbation technique similar to that of Melnikov [1963] for detecting the presence of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations. Our techniques are more general in that they apply to systems undergoing large amplitude excitation at low frequencies and to systems undergoing quasiperiodic excitation. We discuss the dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and the recent results concerning orbits homoclinic to tori obtained by Wiggins [1986] and Meyer and Sell [1986] . We illustrate the method with an example of a parametrically excited pendulum. Additionally, we show how our results are complementary to a result of Arnold [1962] concerning the infinite time behavior of adiabatic invariants in Hamiltonian systems. We illustrate this by using our method to prove the existence of Smale horseshoes for a pendulum whose length undergoes a slow periodic variation.