Existence and Computation of Hyperbolic Trajectories of Aperiodically Time Dependent Vector Fields and their Approximations

In this paper we give sufficient conditions for the existence of hyperbolic trajectories in aperiodically time dependent vector fields. These conditions do not require the a priori introduction of hyperbolicity into the dynamics of the vector field or assumptions of "time scale separation". The hyperbolic trajectory is obtained as a solution of an integral equation over an infinite time interval. We give an expression for the error obtained when the solution is approximated over a finite time interval. Finally, we show how the method can be numerically implemented in a specific example.