Symmetries in the Lorenz-96 model

TL;DR

This paper analyzes the symmetry-induced bifurcation structure of the Lorenz-96 model for negative forcing, revealing patterns of pitchfork bifurcations across different dimensions using equivariant bifurcation theory.

Contribution

It provides a detailed theoretical analysis of bifurcations in the symmetric Lorenz-96 model for negative forcing, extending results to all multiples of certain dimensions.

Findings

Existence of supercritical pitchfork bifurcations in even dimensions.

Multiple bifurcations in dimensions divisible by 4.

Cascade of bifurcations in dimensions of the form 2^q p, with p odd.

Abstract

The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing $F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is $\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$ using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces, that play an important role in this model. We exploit them in order to generalise results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for $F<0$ in specific dimensions $n$: In all even dimensions, the equilibrium $(F,\ldots,F)$ exhibits a supercritical pitchfork bifurcation. In dimensions $n=4k$, $k\in\mathbb{N}$, a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension $n=2^qp$, where $q\in\mathbb{N}\cup\{0\}$ and $p$ is odd, there is a finite cascade of exactly $q$ subsequent pitchfork bifurcations, whose bifurcation values are independent of $n$. This structure is discussed and interpreted in light of the symmetries of the model.