FTLE time-window criterion for bubble-free synchronization in Rössler networks

Establish that, for networks of diffusively coupled chaotic Rössler oscillators, bubble-free synchronization occurs when the averaging window τ_{Amax} that achieves the maximum finite-time transverse Lyapunov amplification satisfies τ_{Amax} < T_th, with T_th equal to the oscillator’s average period T_ave.

Background

The authors introduce a finite-time Lyapunov–based indicator, where the maximum amplification and the corresponding averaging time τ{Amax} sharply change at the bubbling–bubble-free transition. They propose a practical criterion using τ{Amax} and an oscillator-specific threshold T_th and, for Rössler networks, posit T_th equals the average period.

They explicitly formulate this as a conjecture, suggesting a clear, testable condition linking finite-time transverse Lyapunov averaging to bubble suppression.