Characterization of Local Structures of Chaotic Attractors in Terms of Coarse-Grained Local Expansion Rates

Abstract

Chaotic attractors at the bifurcation points of band mergings (or splittings), crises and saddlenode bifurcations have singular local structures which produce coherent large fluctuations of the coarse-grained local expansion rates of nearby orbits. Such local structures are studied in terms of a weighted average Λ(q), (-∞≪q≪∞) of the coarse-grained local expansion rates along the local unstable manifold with a q-dependent weight. By taking invertible two-dimensional and noninvertible one-dimensional maps, it is shown that, as q is varied, Λ(q) exhibits discontinuous phase transitions at discrete values of q;qα, qβ,…. Three types of such phase transitions are discussed. One is that caused by the homoclinic tangencies with qα=2.0. The second is that due to the accumulation of homoclinic tangency points at unstable periodic points with qβ≃-0.8. The third is that due to the intermittent hopping motions between two repellers with qγ=1.0. Different phases of the phase transitions represent different local structures. Thus the q-weighted average Λ(q) turns out to provide a useful means for characterizing singular local structures of chaotic attractors.