# Travelling waves and their bifurcations in the Lorenz-96 model

## TL;DR

This paper analyzes the Lorenz-96 model's bifurcations and traveling waves, revealing supercritical Hopf bifurcations, quasi-periodic attractors, and routes to chaos through analytical and numerical methods.

## Contribution

It provides explicit formulas for bifurcation analysis and demonstrates the role of Hopf-Hopf bifurcations as organizing centers in the Lorenz-96 model.

## Key findings

- First Hopf bifurcation is always supercritical.
- Traveling wave solutions emerge at bifurcation.
- Routes to chaos vary with no clear pattern as dimension increases.

## Abstract

In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter $F$ are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension $n\geq4$. Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for $F>0$ is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave. Furthermore, by unfolding the codimension two Hopf-Hopf bifurcation it is shown to act as an organising centre, explaining dynamics such as quasi-periodic attractors and multistability, which are observed in the original Lorenz-96 model. Finally, the region of parameter values beyond the first Hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and Lyapunov exponents. The observed routes to chaos are various but without clear pattern as $n\rightarrow\infty$.