In this paper we show, both analytically and experimentally, that the Rössler system synchronization is either asymptotically stable or orbitally stable within a wide range of the system key parameters. In the meantime, we provide some simple sufficient conditions for synchronization stabilities of the Rössler system in a general situation. Our computer simulation shows that the type of stability of the synchronization is very sensitive to the initial values of the two (drive and response) Rössler systems, especially for higher-periodic synchronizing trajectories, which is believed to be a fundamental characteristic of chaotic synchronization that preserves the extreme sensitivity to initial conditions of chaotic systems.
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