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

Chaotic attractors at the bifurcation points of band mergings (or splittings), crises and saddle· node 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 A(q), (-00< q< (0) of the coarse·grained local expansion rates along the local unstable manifold with a q·dependent weight. By taking invertible two·dimensional and noninverti· ble one·dimensional maps, it is shown that, as q is varied, A(q) exhibits discontinuous phase transi· tions at discrete values of q; qa, qp, .... Three types of such phase transitions are discussed. One is that caused by the homoc1inic tangencies with qa=2.0. The second is that due to the accumulation of homoc1inic tangency points at unstable periodic points with qp"'" -0.8. The third is that due to the intermittent hopping motions between two repellers with qT=LO. Different phases of the phase transitions represent different local structures. Thus the q-weighted average A(q) turns out to provide a useful means for characterizing singular local structures of chaotic attractors.