An operator methodology for the global dynamic analysis of stochastic nonlinear systems

In a global dynamic analysis, the coexisting attractors and their basins are the main tools to understand the system behavior and safety. However, both basins and attractors can be drastically influenced by uncertainties. The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors. Accordingly, analytical and numerical tools for calculation of nondeterministic global structures, namely attractors and basins, are proposed. First, based on the definition of the Perron-Frobenius, Koopman and Foias linear operators, a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases. In this context, the stochastic basins of attraction and attractors' distributions replace the usual basin and attractor concepts. Then, numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method. Sample results of the methodology are presented for a canonical dynamical system.