Characterization of Local Structures of Chaotic Attractors in Terms of Coarse-Grained Local Expansion Rates: General and Mathematical Physics

概要

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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; q_α q_β…. Three types of such phase transitions are discussed. One is that caused by the homoclinic tangencies with q_α =2.0. The second is that due to the accumulation of homoclinic tangency points at unstable periodic points with q_β≃-0.8. The third is that due to the intermittent hopping motions between two repellers with q_γ =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.
理論物理学刊行会の論文
1988-11-25