Periodic perturbations of constrained motion problems on a class of implicitly defined manifolds

TL;DR

This paper investigates the existence of periodic solutions in constrained dynamical systems on manifolds defined by smooth equations, using degree theory to analyze forced oscillations under perturbations.

Contribution

It introduces degree-theoretic conditions for periodic solutions in constrained systems and applies these to differential-algebraic equations with periodic forcing.

Findings

Established conditions for existence of periodic solutions

Applied theory to differential-algebraic equations

Provided a framework for analyzing forced oscillations on manifolds

Abstract

We study forced oscillations on differentiable manifolds which are globally defined as the zero set of appropriate smooth maps in some Euclidean spaces. Given a T-periodic perturbative forcing field, we consider the two different scenarios of a nontrivial unperturbed force field and of perturbation of the zero field. We provide simple, degree-theoretic conditions for the existence of branches of T-periodic solutions. We apply our construction to a class of second order Differential-Algebraic Equations.