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Expansivity of Nonsmooth Functional Differential Equations

Expansivity of a dynamical system, in which all trajectories that remain within a certain threshold of each other must be identical, is a fundamental characteristic of systems with complicated dynamical behaviour and, indeed, can be considered as a basic mechanism for producing such behaviour within a compact invariant set. In many expansive systems, w x especially those involving a hyperbolic or semi-hyperbolic structure 1, 3]5 , there is in fact an exponential rate of separation of distinct trajectories in the region of interest. Explicit knowledge of the rates of separation is useful for numerical calculations and shadowing arguments. In this paper sufficient conditions are presented for the exponential expansivity of a discrete-time dynamical system that is generated by a continuous mapping of a Banach space into itself. This mapping need not be differentiable or invertible and the subset under consideration need not be invariant. The main result of the paper is to show that these conditions are satisfied by the shift mapping of a nonlinear functional differential equation formed by sufficiently small Lipschitz perturbations of a linear functional differential equation with a hyperbolic equilibrium point. In