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 mechanic!~ of irreversible processes. The broken timereversal symmetry is brought about by orbital instability of chaos.
The probability density distribution is studied analytically and by Monte Carlo simulations for a periodically driven chemical bistable system, described by a master equation, for the case of low-frequency driving. The quasistationary distribution about the stable states is well approximated by the solution of the master equation in the eikonal approximation for large volumes of the system. For a one-component system both the exponent and the prefactor of the steady distribution are obtained in explicit form, for an arbitrary strength of the driving and for an arbitrary interrelation between the frequency of the driving and the probabilities of transitions between the stable states. The results of the simulations are in good agreement with analytical results. We demonstrate the onset of stochastic resonance for the driving frequency close to the probabilities of fluctuational transitions between the states.
Nonlinear dynamics of coupled FitzHugh-Nagumo neurons subject to independent noise is analyzed. A kind of self-sustained global oscillation with almost synchronous firing is generated by array-enhanced coherence resonance. Further, forced dynamics of the self-sustained global oscillation stimulated by sinusoidal input is analyzed and classified as synchronized, quasiperiodic, and chaotic responses just like the forced oscillations in nerve membranes observed by in vitro experiments with squid giant axons. Possible physiological importance of such forced oscillations is also discussed.
1065The repeated sticking of a chaotic orbit to critical tori with an inverse· power distribution of sticking times is shown to produce a universal spectrum of expansion rates A of nearby orbits with a linearity with zero slope for O
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