An adaptative phase-space discretization strategy for the global analysis of stochastic nonlinear dynamical systems with competing attractors considering parameter uncertainty or 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 and densities. 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 converging 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. Finally, a Monte Carlo procedure is conducted for the stochastic case to construct the proper transfer operator. The present formulation is employed in Part II to investigate the global dynamics of two archetypal nonlinear oscillators.

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This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron-Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modelled using a nonlinear Bernoulli-Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model (mdof) model, an accurate single degree of freedom (sdof) reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviours. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors` distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behaviour. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular MEMS, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

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This paper explores the influence of uncertainties and noise on the global dynamics of nonlinear systems, with emphasis on the basins and attractors’ topology and on the dynamic integrity of coexisting solutions. For this, the adaptive phase-space discretization strategy proposed in Part I [1] is employed. Two well-known archetypal oscillators found in engineering applications, the Helmholtz and Duffing oscillators under harmonic or parametric excitation with uncertain parameters or added load noise, are studied. The results demonstrate the adaptive capability of the proposed method, increasing the resolution without increasing the computational cost. Mean basins of attraction and attractors’ distributions are obtained. The time-dependency of stochastic responses is demonstrated, with long-transients influencing the short-term behavior. Finally, the effect of uncertainties and noise on the basins areas, attractors distributions and basin boundaries is discussed, which can be used to evaluate the dynamic integrity of the competing basins.

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