Local rough stable manifolds without vanishing-derivative assumption

Establish the existence of local rough stable invariant manifolds near the coherent equilibrium for the rough Kuramoto model with identical natural frequencies and symmetric connected coupling (Hypotheses A and B), driven by a rough noise W satisfying Hypothesis H_W and a rotationally invariant noise coefficient G ∈ C_b^3 that satisfies the cancellation condition ∑_{i=1}^N G_{ij}(θ)=0 for all j (Hypothesis H_G+), without imposing the vanishing-derivative assumption DG(0)=D^2G(0)=0. In particular, prove that a local stable invariant manifold exists around the synchronized state (θ_1=⋯=θ_N) even when DG(0) and D^2G(0) are nonzero.

Background

In Subsection "Deterministic Synchronization for particular noise," the authors show that if the noise coefficient G additionally satisfies the algebraic cancellation condition (H_G+) and if DG(0)=D^2G(0)=0, then local rough stable invariant manifolds around coherent equilibria can be characterized almost surely. This relies on existing invariant manifold theory for rough differential equations that assumes vanishing derivatives of the noise at the equilibrium.

They conjecture that the derivative-vanishing condition is not essential for the existence of such local manifolds in their Kuramoto setting. Related work based on the multiplicative ergodic theorem can yield stable manifolds under different assumptions but does not directly provide the desired graph structure; thus, removing the DG(0)=D^2G(0)=0 requirement remains unresolved in this framework.