Computation of Lyapunov-type numbers for invariant curves of planar maps

Invariant manifolds play an important role in the study of dynamical systems, and it is often crucial to know when they persist under small perturbations of the system. Sufficient conditions for persistence have been formulated in the literature in terms of so-called Lyapunov-type numbers, which measure and compare attractivity rates. The numerical evaluations of these numbers in a simple setting is the subject of this paper. We consider planar diffeomorphisms depending on a parameter; the diffeomorphisms have invariant curves which deform as the parameter changes. The Lyapunov-type numbers are then monitored as functions of parameter and are related to the dynamics on the invariant curves. As an example, we consider the delayed logistic map. We also describe how the invariant curves have been computed.