Bifurcation of closed orbits from equilibria of Newtonian systems with Coriolis forces

TL;DR

This paper studies the existence and bifurcation of periodic orbits from equilibria in Newtonian systems with Coriolis forces, using both local and global mathematical tools, and applies findings to celestial mechanics.

Contribution

It introduces a combined approach using Lyapunov's center theorem and equivariant degree theory to analyze bifurcations, including degenerate cases, in systems with Coriolis forces.

Findings

Existence of multiple bifurcating periodic orbits from equilibria.

Application to celestial mechanics confirms at least seven orbit branches.

Global and local methods complement each other in bifurcation analysis.

Abstract

We consider autonomous Newtonian systems with Coriolis forces in two and three dimensions and study the existence of branches of periodic orbits emanating from equilibria. We investigate both degenerate and nondegenerate situations. While Lyapunov's center theorem applies locally in the nondegenerate, nonresonant context, equivariant degree theory provides a global answer which is significant also in some degenerate cases. We apply our abstract results to a problem from Celestial Mechanics. More precisely, in the three-dimensional version of the Restricted Triangular Four Body Problem with possibly different primaries our results show the existence of at least seven branches of periodic orbits emanating from the stationary points.