Global analysis of stochastic and parametric uncertainty in nonlinear dynamical systems: adaptative phase-space discretization strategy, with application to Helmholtz oscillator

An adaptative phase-space discretization strategy for the global analysis of stochastic nonlinear dynamical systems with competing attractors considering parameter uncertainty and noise is proposed. The strategy is based on the classical Ulam method. The appropriate transfer operators for a given dynamical system are derived and applied to obtain and refine the basins of attraction boundaries and attractors distributions. A review of the main concepts of parameter uncertainty and stochasticity from a global dynamics perspective is given, and the necessary modifications to the Ulam method are addressed. The stochastic basin of attraction definition here used replaces the usual basin concept. It quantifies the probability of the response associated with a given set of initial conditions to converge to a particular attractor. The phase-space dimension is augmented to include the extra dimensions associated with the parameter space for the case of parameter uncertainty, being a function of the number of uncertain parameters. The expanded space is discretized, resulting in a collection of transfer operators that enable obtaining the required statistics. A Monte Carlo procedure is conducted for the stochastic case to construct the proper transfer operator. An archetypal nonlinear oscillator with noise and uncertainty is investigated in-depth through the proposed strategy, showing significative computational cost reduction.