Statistical Mechanics of Dynamical Systems : Complex Dynamics in Nonlinear Systems

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A statistical-mechanical formalism of chaos based on the geometry of invariant sets in phase space is discussed to show that chaotic dynamical systems can be treated by a formalism analogous to that of thermodynamic systems if one takes a relevant coarse-grained quantity, but their statistical laws are quite different from those of thermodynamic systems. This is a generalization of statistical mechanics for dealing with dissipative and hamiltonian (i. e., conservative) dynamical systems of a few degrees of freedom. Thus the sum of the local expansion rate of nearby orbits along a relevant orbit over a long but finite time has been introduced in order to describe and characterize (1) a drastic change of the structure of a chaotic attractor at a bifurcation and anomalous phenomena associated, (2) a critical scaling of chaos in the neighborhood of a critical point for the bifurcation to a nonchaotic state, and a self-similar temporal structure of a critical orbit on the critical 2^∞ attractor and the critical golden tori without mixing, (3) the critical KAM torus, diffusion and repeated sticking of a chaotic orbit to a critical torus in hamiltonian systems. Here a q-phase transition, analogous to the ferromagnetic phase transition, plays an important role. They are illustrated numerically and theoretically by treating the driven damped pendulum, the driven Duffing equation, the Henon map, and the dissipative and conservative standard maps. This description of chaos breaks the time-reversal symmetry of hamiltonian dynamical laws analogously to statistical mechanics of irreversible processes. The broken time-reversal symmetry is brought about by orbital instability of chaos.
理論物理学刊行会の論文
1990-03-28

Statistical Mechanics of Dynamical Systems : Complex Dynamics in Nonlinear Systems

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