Study on the Onset and Structures of Chaos from the Standpoint of Statistical Physics

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The purpose of this project is to reveal characteristic strctures of chaos observed ubiquitously in dissipative dynamical systems and to obtain a clue for constructing the statistical mechanics of chaos from the first principle. We take the standpoint that an essential mechanism of chaos in dissipative systems will be found in low-dimensional discrete time dynamical systems. Thus chaos generated by one- and two-dimensional maps are considered. We started with chaos near its onset point and the characteristic structures of power spectra of orbits in this chaos have been found. Then we proceeded to off-boundary chaos and its characterisic properties were investigated in terms of two quantities: one is the spectrum of singularities characterizing multifractal structures of measures on strange attractors and the other the spectrun of fluctuation of local expansion rates of chaotic orbits. The main results are as follows: 1. The power spectra of periodic chaos which emerges via period-doubling bifurcations show characteristic structures governed by the universal recursion relations derived on the basis of similarities in the process of band-splitting bifurcations of this chaos. 2. The power spectra of intermittent chaos are characterized by a series of peaks and the power low for their envelope, the statistical properties of the duration of laminar motion and the jumps in amplitude and phase by bursts being important factors. 3. The characteristic relations between the spectrum of singularities and the fluctuation spectrum of local expansion rates are obtained for certain kinds of chaos in hyperbolic dynamical systems. 4. Phase transitions associated with anomalous flucturations of local expansion rates are found at bifurcation points of chaos on the basis of the formalism similar to the usual statistical mechanics. 5. Critical exponents and scalling functions near the phase transition point stated in 4 are obtained for certain systems.