Exploring the qualitative behavior of uncertain dynamical systems

Abstract

The global behavior of uncertain nonlinear systems by means of the cell mapping method is investigated. The systems under consideration are governed by ordinary differential equations where uncertainty is characterized by a bounded stationary noise process. In order to make cell mapping methods applicable to this particular situation generalizations of the theory become necessary. The numerical approach is based on a reformulation of the dynamics in terms of a Markov process whose qualitative behavior is analyzed. We demonstrate the usefulness of the presented scheme in the engineering field while addressing an example problem from technical mechanics.

Modeling of nonlinear dynamical systems based on deterministic learning and structural stability

“…As a fundamental property of nonlinear dynamical systems, structural stability provides a justification for applying the qualitative theory of dynamical systems to analyze physical systems [33]. Unlike Lyapunov stability [34], which studies the influence of perturbations of the initial condition on dynamical behaviors of a dynamical system itself, structural stability reveals that the qualitative dynamical behaviors (i.e.…”

Probabilistic approach to large amplitude ship rolling in random seas

“…The basics of this method were developed by Hsu [17,18]. Recently, an extension of this method was proposed [19] which is appropriate for the global analysis of non-linear dynamical systems under stochastic excitations. However, the high computing time required by the cell mapping method allows yet the analysis of lower dimensional systems only.…”