Detecting invariant manifolds, attractors, and generalized KAM tori in aperiodically forced mechanical systems

We show how the recently developed theory of geodesic transport barriers for fluid flows can be used to uncover key invariant manifolds in externally forced, one-degree-of-freedom mechanical systems. Specifically, invariant sets in such systems turn out to be shadowed by least-stretching geodesics of the Cauchy–Green strain tensor computed from the flow map of the forced mechanical system. This approach enables the finite-time visualization of generalized stable and unstable manifolds, attractors and generalized KAM curves under arbitrary forcing, when Poincaré maps are not available. We illustrate these results by detailed visualizations of the key finite-time invariant sets of conservatively and dissipatively forced Duffing oscillators. A. Hadjighasem Department of Mechanical Engineering, McGill University, 817 Sherbrooke Ave. West, Montreal, Quebec H3A 2K6, Canada e-mail: [email protected] M. Farazmand · G. Haller (!) Institute for Mechanical Systems, ETH Zürich, Tannenstrasse 3, 8092 Zürich, Switzerland e-mail: [email protected] M. Farazmand Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland e-mail: [email protected]