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 [Formula: see text] and the dimension [Formula: see text] as parameters and is [Formula: see text]-equivariant. In this paper, we unravel its dynamics for [Formula: see text] using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces that play an important role in this model. We exploit them in order to generalize 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 [Formula: see text] in specific dimensions [Formula: see text]: In all even dimensions, the equilibrium [Formula: see text] exhibits a supercritical pitchfork bifurcation. In dimensions [Formula: see text], [Formula: see text], a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension [Formula: see text], where [Formula: see text] and [Formula: see text] is odd, there is a finite cascade of exactly [Formula: see text] subsequent pitchfork bifurcations, whose bifurcation values are independent of [Formula: see text]. This structure is discussed and interpreted in light of the symmetries of the model.