Localized synchrony patterns in weakly coupled bistable oscillator systems

The paper proves, for dissipative coupling c=1, the existence of two unique smooth snaking branches Γ^{on/off}_ε(N) inside a neighborhood of the ε=0 skeleton Γ^δ_0(N), with C^0 convergence as ε→0, Cm away from µ∈{0,1}, and generic folds near µ=0 and µ=1; see the definition of Γ^δ_0(N) and Theorem 2.1, together with the amplitude–phase decomposition (3.1)–(3.2), the core/far-field elimination (3.4)–(3.5), and the invertibility in Lemma 3.3 and the fold analyses in §3.3–§3.4 . The model’s solution reaches the same conclusions via an equivalent but differently organized IFT/Lyapunov–Schmidt route (using Gershgorin for uniform invertibility and a one-dimensional normal form near the soft mode), so the conclusions agree while the proof techniques differ.