# Periodic orbits in nonlinear wave equations on networks

## TL;DR

This paper analyzes nonlinear wave equations on networks, identifying conditions for periodic orbits based on graph modes, and investigates their stability through Floquet analysis, with implications for coupled mechanical systems.

## Contribution

It introduces a method to identify nonlinear periodic orbits from graph modes and analyzes their stability, distinguishing between modes with and without soft nodes.

## Key findings

- Goldstone mode is usually stable.
- Bivalent mode is always unstable.
- Modes with a single eigenvalue are unstable below a threshold.

## Abstract

We consider a cubic nonlinear wave equation on a network and show that inspecting the normal modes of the graph, we can immediately identify which ones extend into nonlinear periodic orbits. Two main classes of nonlinear periodic orbits exist: modes without soft nodes and others. For the former which are the Goldstone and the bivalent modes, the linearized equations decouple. A Floquet analysis was conducted systematically for chains; it indicates that the Goldstone mode is usually stable and the bivalent mode is always unstable. The linearized equations for the second type of modes are coupled, they indicate which modes will be excited when the orbit destabilizes. Numerical results for the second class show that modes with a single eigenvalue are unstable below a treshold amplitude. Conversely, modes with multiple eigenvalues seem always unstable. This study could be applied to coupled mechanical systems.