Localized Instability and Attraction along Invariant Manifolds

Abstract. We derive a simple criterion for transverse instabilities along a general invariant manifold of a multidimensional dynamical system. The criterion requires an appropriately defined normal infinitesimal Lyapunov exponent (NILE) to be positive over regions of transverse instability on the manifold. Unlike classic Lyapunov-type numbers in the theory of normally hyperbolic invariant manifolds, the NILE can be computed analytically in applications. This enables us to locate, for example, regions of transient jumping along an invariant manifold that is otherwise globally attracting. To illustrate our results, we determine the locations of intermittent instabilities in bubble motion past a cylinder, in predator-prey interactions, and in soft-stiff structural systems. Key words. normal stability of invariant manifolds, slow manifold reduction, local instabilities along invariant manifolds, infinitesimal Lyapunov exponents AMS subject classifications. 37D05, 37D10, 37D25 DOI. 10.1137/...

• Type: Journal
• Year: 2010
• Where: SIAMADS
• Authors: George Haller, Themistoklis Sapsis
• Keywords: General Invariant Manifold, Infinitesimal Lyapunov, Invariant Manifolds, SIAMADS 2010