Symmetries in the Lorenz-96 model

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 ∈ R and the dimension n ∈ N as parameters and is Zn-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 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 F < 0 in specific dimensions n: In all even dimensions, the equilibrium (F, . . . , F) exhibits a supercritical pitchfork bifurcation. In dimensions n = 4k, k ∈ N, a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one.
Furthermore, numerical observations reveal that in dimension n = 2qp, where q ∈ N ∪ {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.