Global persistence of Lyapunov-subcenter-manifolds as spectral submanifolds under dissipative perturbations

TL;DR

This paper proves that Lyapunov Subcenter Manifolds, which are key in understanding nonlinear vibrations, persist under small dissipative perturbations and provides detailed mathematical analysis of their dependence on dissipation.

Contribution

The paper establishes rigorous conditions for the persistence of Lyapunov Subcenter Manifolds under dissipation and analyzes their dependence on the dissipation parameter.

Findings

Lyapunov Subcenter Manifolds persist under small dissipation.

Manifolds are real analytic in dissipation parameter away from zero.

Explicit asymptotic expansions in powers of dissipation are constructed.

Abstract

For a nondegenerate analytic system with a conserved quantity, a classic result by Lyapunov guarantees the existence of an analytic manifold of periodic orbits tangent to any two-dimensional, elliptic eigenspace of a fixed point satisfying nonresonance conditions. These two dimensional manifolds are referred as Lyapunov Subcenter Manifolds (LSM). Numerical and experimental observations in the nonlinear vibrations literature suggest that LSM's often persist under autonomous, dissipative perturbations. These perturbed manifolds are useful since they provide information of the asymptotics of the convergence to equilibrium. In this paper, we formulate and prove precise mathematical results on the persistence of LSM under dissipation. We show that, under mild non-degeneracy conditions on the perturbation, for small enough dissipation, there are analytic invariant manifolds of the perturbed system that approximate (in the analytic sense) the LSM in a fixed neighborhood. We also provide examples that show that some non-degeneracy conditions on the perturbations are needed for the results to hold true. We also study the dependence of the manifolds on the dissipation parameter. If $\varepsilon$ is the dissipation parameter, we show that the manifolds are real analytic in $(-\varepsilon_0, \varepsilon_0) \setminus \{0\} $ and $C^\infty $ in $(-\varepsilon_0, \varepsilon_0)$. We construct explicit asymptotic expansions in powers of $\varepsilon$ (which presumably do not converge). Finally, we present applications of our results to several mechanical systems.