Predict the threshold C_th for bubble-free synchronization from cascaded UPO amplification

Determine, via a nonlinear stability analysis tailored to diffusively coupled networks of chaotic Rössler oscillators (with Laplacian coupling across all state variables), the threshold value C_th of the cascaded amplification factor C = exp{∑_i τ_i^{UPO} λ_i^{⊥,UPO}} that delineates bubble-free synchronization from bubbling, where the sum runs over all transversely unstable unstable periodic orbits on the synchronization manifold, τ_i^{UPO} = 1/λ_i^{∥,UPO} is the typical residence time near the i-th orbit, and λ_i^{⊥,UPO} and λ_i^{∥,UPO} are the transverse and longitudinal Lyapunov exponents of that orbit, respectively.

Background

In the analysis of bubbling onset, the authors introduce a cascaded amplification factor C that accumulates the net exponential growth produced when trajectories successively visit transversely unstable unstable periodic orbits (UPOs) on the synchronization manifold. This is a worst-case “perfect storm” scenario in which the trajectory visits all transversely unstable UPOs.

They posit that bubble-free synchronization requires C < C_th, but emphasize that obtaining C_th demands a nonlinear stability analysis depending on oscillator dynamics. Empirically, they estimate C_th ≈ 3 for their Rössler network at the bubble–bubble-free transition, and suggest a normal form analysis could yield a principled prediction of C_th.