Singular Perturbation Methods for Nonlinear Dynamical Systems and Waves

摘要

Progress has been made on understanding the complex behavior of physicalprocesses described by nonlinear ordinary and partial differential equations through the use of singular perturbation methods. Modulation equations for the amplitude and phase of dissipatively perturbed strongly nonlinear oscillators and traveling waves have been derived from the action equation using the usual method of multiple scales. Equivalent results have been obtained using the method of averaging developed for the first time for a nonlinear partial differential equation, the Klein-Gordon equation, describing dispersive waves. In another study, Whitham's averaged Lagrangian principle has been generalized to account for arbitrary perturbations of the initial conditions. In other work, Bourland and Haberman analyzed the slow crossing of an uperturbed homoclinic orbit (separatrix) for dynamical systems. Solutions in the neighborhood of the separatrix are matched to the nonlinear slowly varying oscillations, resulting in the determination of accurate analytic formulas for the boundaries of the basin of attraction and connection formulas across the separatrix for the amplitude and phase. Under current investigation are generalizations of the slow crossing of a separatrix to arbitrary Hamiltonian systems and to nonchaotic situations in which small periodic forcing causes the existence of an infinite sequence of resonance layers that coalesce on the separatrix.