The use of manifold tangencies to predict orbits, bifurcations and estimate escape in driven systems

Oscillatory solutions for the parametrically excited pendulum are considered, and comparison is made between the pendulum and systems that permit escape from a symmetric potential well. Unstable and stable orbits are identified and located in control space by considering the horseshoe formed by the invariant manifolds of the hilltop saddles at 0 = ±n, and employing methods of symbolic dynamics. Observing the tangencies in the manifolds from these saddles allows considerable detail regarding the bifurcational structure of the system to be determined without recourse to lengthy simulations. Estimation of changes in the Birkhoff signature additionally permits estimates of escape for systems with symmetric and asymmetric potential wells. These numerical/analytical results can prove to be very useful in estimating global dynamics which may in the future be employed in experimental systems.