Computation of Lyapunov measure for almost everywhere stability

In our recent paper (Vaidya, 2006) Lyapunov measure is introduced as a new tool for verifying almost everywhere stability of an invariant set in a nonlinear dynamical system or continuous mapping. It is shown that for almost everywhere stable system explicit formula for the Lyapunov measure can be obtained as a infinite series or as a resolvent of stochastic linear operator. This paper focus on the computation aspects of the Lyapunov measure. Methods for computing these Lyapunov measures are presented based upon set-oriented numerical approaches, which are used for the finite dimensional approximation of the linear operator. Stability results for the finite dimensional approximation of the linear operator are presented. The stability in finite dimensional space results in further weaker notion of stability which in this paper is referred to as coarse stability