Travelling waves and their bifurcations in the Lorenz-96 model

Highlights: • The existence of Hopf and Hopf-Hopf bifurcations is proven for all $n\ge 4$. • An analytical formula is derived for the first Lyapunov coefficient of the Hopf bifurcation. • Periodic attractors have the physical interpretation of travelling waves. • The Hopf-Hopf bifurcation acts as an organising center and leads to multistability. • The dynamics beyond the first bifurcation depends on $n$, but without clear pattern. 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 $n\ge 4$. 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 $n\to \infty$.