Multiple Periodic Solutions for $\Gamma$-symmetric Newtonian Systems

TL;DR

This paper develops methods to compute equivariant degrees for $ ext{Gamma}$-symmetric Newtonian systems, enabling the identification of multiple periodic solutions with various symmetries.

Contribution

It introduces effective computational techniques for the Euler ring and degrees in $ ext{Gamma} imes O(2)$-equivariant settings, facilitating the analysis of symmetric Newtonian systems.

Findings

Multiple periodic solutions are proven to exist in systems with various symmetries.

Explicit calculations of topological invariants for specific systems are provided.

The methods enable effective computation of degrees using Euler and Burnside rings.

Abstract

The existence of periodic solutions in $\Gamma$-symmetric Newtonian systems $\ddot{x}=-\nabla f(x)$ can be effectively studied by means of the $(\Gamma\times O(2))$-equivariant gradient degree with values in the Euler ring $U(\Gamma\times O(2))$. In this paper, we show that in the case of $\Gamma$ being a finite group, the Euler ring $U(\Gamma\times O(2))$ and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring $A(\Gamma\times O(2))$ and the reduced $(\Gamma\times O(2))$-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.