Hyperbolic quasiperiodic solutions of U-monotone systems on Riemannian manifolds

TL;DR

This paper establishes conditions for the existence, hyperbolicity, and uniqueness of quasiperiodic solutions in U-monotone systems on Riemannian manifolds, using topological and convex analysis methods, with applications to charged particle motion.

Contribution

It introduces the concept of U-monotonicity for second order non-autonomous systems on Riemannian manifolds and derives conditions ensuring quasiperiodic solutions with stability and bounds.

Findings

Existence of bounded quasiperiodic solutions under specific conditions.

Proved hyperbolicity and uniqueness of solutions.

Applied results to charged particle dynamics on a sphere.

Abstract

We consider a second order non-autonomous system which can be interpreted as the Newtonian equation of motion on a Riemannian manifold under the action of time-quasiperiodic force field. The problem is to find conditions which ensures: (a) the existence of a solution taking values in a given bounded domain of configuration space and possessing a bounded derivative; (b) the hyperbolicity of such a solution; (c) the uniqueness and, as a consequence, the quasiperiodicity of such a solution. Our approach exploits ideas of Wa\.zewski topological principle. The required conditions are formulated in terms of an auxiliary convex function $U$. We use this function to establish the Landau type inequality for the derivative of solution, as well as to introduce the notion of U-monotonicity for the system. The U-monotonicity property of the system implies the uniqueness and the quasiperiodicity of its bounded solution. We also find the bounds for magnitude of perturbations which do not destroy the quasiperiodic solution. The results obtained are applied to study the motion of a charged particle on a unite sphere under the action of time-quasiperiodic electric and magnetic fields.