Statistical dynamics of sine-square map

In this paper we study (i) the variation of local Lyapunov exponent, (ii) characterization of chaotic attractor at bifurcations using the variance σn(q) of fluctuations of coarse-grained local expansion rates of nearby orbits and (iii) characterization of weak and strong chaos in a sine-square map which describes the dynamics of the liquid crystal hybrid optical bistable device. The standard deviation of local Lyapunov exponent λ(X,L) calculated after every L time steps and Allan variance are found to approach zero in the limit L → ∞ as L−α. For all chaotic attractors of the map the σn(q) versus q plot exhibits a peak at q = qα. We show that additional peaks, however, occur only for the attractors just before and just after the bifurcations. We investigate the characteristics of the probability distributions of a k-step difference quantity ΔXk=Xi+k−Xi. We show that a nonstationary probability distribution occurs for weak chaos and a stationary distribution occurs for strong chaos.