Nonlinear Systems With Multiplicative and Additive Perturbation Under State Space Constraints

Realistic models of mechanical systems often depend on various parameters, such as controlled inputs, material constants, tunable parameters, and uncertainties. Uncertain parameters can be (time varying) deterministic perturbations or stochastic excitations whose influence on the system depends on the perturbation dynamics (multiplicative or additive), the perturbation range, and its statistics in the stochastic case. For a given operating region of the system, i.e. for a set of state space constraints, the behavior of the system within this region depends strongly on the type of perturbation dynamics and on its range. We present some basic theory for additively and multiplicatively perturbed systems, where the uncertainty can be a family of time varying functions, or a Markov diffusion process. The uncertainty range plays the role of a bifurcation parameter and determines concepts like discontinuities of control sets and supports of invariant measures, stability radii, and invariance radii with respect to the constraint set. It turns out that in many instances the stochastic and the deterministic bifurcation scenarios agree, and the cases in which they differ are related to a nonuniform behavior of the stochastically perturbed system. The example of a model of ship roll motion is treated in detail, revealing some of the fundamental agreements and disagreements of the two bifurcation scenarios.