Degree of synchronization of noisy maps on the circle

We consider two systems of nearly identical mappings on the circle, each having a single fixed point and additive noise. The noise moves each system between stable and unstable regions of the map, causing the separation of the systems to fluctuate. The geometric distribution of separations is characterized by a distribution exponent, which describes the degree of synchronization between the two systems. The distribution exponent is related to the conditional Lyapunov exponent, the more usual way of characterizing synchronization. The relation between the distribution and Lyapunov exponents is determined near the threshold of synchronization, where we prove that the two exponents change sign together. Fluctuations of separation that appear in systems driven by noise also appear in systems driven by chaos so that many of the methods may be useful for analyzing chaotic synchronization. {} 1996 The American Physical Society.