Unstable dimension variability and synchronization of chaotic systems

The nonhyperbolic structure of synchronization dynamics is investigated in this work. We argue analytically and confirm numerically that the chaotic dynamics on the synchronization manifold exhibits an unstable dimension variability, which is an extreme form of nonhyperbolicity. We analyze the dynamics in the synchronization manifold and in its transversal direction, where a tonguelike structure is formed, through a system of two coupled chaotic maps. The unstable dimension variability is revealed in the statistical distribution of the finite-time transversal Lyapunov exponent, having both negative and positive values. We also point out that unstable dimension variability is a cause of severe modeling difficulty.