Coexistence of synchronization manifolds in networks of oscillators with rotation symmetry

The master stability function (MSF) formalism provides a powerful framework for analyzing the stability of the synchronization manifold (SM) in networks of coupled oscillators, yielding a necessary condition for synchronization. In this work, we show that the MSF approach naturally extends to dynamical systems possessing rotation symmetry. As a representative case, we consider the Lorenz system, which is invariant under the transformation (x,y,z)↦(-x,-y,z). This corresponds to a rotation symmetry in which z is the invariant variable. We demonstrate that coupling oscillators through an invariant variable preserves the validity of the MSF formalism and leads to the coexistence of multiple SMs, all of which are governed by the same MSF for stability determination. These results highlight how some symmetries can lead to the emergence of multiple SMs and extend the range of collective behaviors that can be analyzed within the MSF framework.