Symmetries in the Lorenz-96 model

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 $(F,\dots ,F)$ 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 $n=2^{q}p$, 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.

31 pages, 9 figures and 3 tables

References in corpus (2)

• Structure of characteristic Lyapunov vectors in spatiotemporal chaos
• Travelling waves and their bifurcations in the Lorenz-96 model

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