Invariant manifolds describing the dynamics of a hyperbolic–parabolic equation from nonlinear viscoelasticity

Abstract

The governing equations for a collection of dynamical problems for heavy rigid attachments carried by light, deformable, nonlinearly viscoelastic bodies are studied. These equations are a discretization of a nonlinear hyperbolic–parabolic partial differential equation coupled to a dynamical boundary condition. A small parameter measuring the ratio of the mass of the deformable body to the mass of the rigid attachment is introduced, and geometric singular perturbation theory is applied to reduce the dynamics to the dynamics of the slow system. Fenichel theory is then applied to the regular perturbation of the slow system to prove the existence of a low-dimensional invariant manifold within the dynamics of the high-dimensional discretization.