Formal Methods for Computing Hyperbolic Invariant Sets for Nonlinear Systems

Hyperbolicity is a cornerstone of nonlinear dynamical systems theory. Hyperbolic dynamics are characterized by the presence of expanding and contracting directions for the derivative along the trajectories of the system. Hyperbolic dynamical systems enjoy many interesting properties like structural stability, ergodicity, transitivity, etc. In this letter, we describe a hybrid systems framework to compute invariant sets with a hyperbolic structure for a given dynamical system. The method relies on an abstraction (also known as symbolic image or bisimulation) of the state space of the system, and on path-complete “Lyapunov-like” techniques to compute the expanding and contracting directions for the derivative along the trajectories of the system. The method is illustrated on a numerical example: the Ikeda map for which an invariant set with hyperbolic structure is computed using the framework.