Analyzing the synchronization of Rössler systems - When trigger-and-reinject is equally important as the spiral motion

The study of the synchronous stability of two coupled Rössler attractors sometimes can be effectively described by approximating the trajectory on the attractor as a planar outward spiral. We show that this is true only when one is dealing with the spiral-type attractor. But when the equally important funnel-type attractor is encountered, a properly constructed time-weighted average must be used to yield a prediction that agrees well with numerical results. We also show analytically how the separation vector evolves in time, and demonstrate why this study matters when one tries to perform the time-weighted average.