Statistical-thermodynamic approach to a chaotic dynamical system: exactly solvable examples. (English) Zbl 1083.37513
Summary: The static and dynamic properties of a chaotic attractor of a two-dimensional map are studied, which belongs to a particular class of piecewise continuous invertible maps. Coverings of a natural size to cover the attractor are introduced, so that the microscopic information of the attractor is written on each box composing the cover. The statistical thermodynamics of the scaling indices and the size indices of the boxes is formulated. Analytic forms of the free energy functions of the scaling indices and the size indices of the boxes are obtained for examples of a hyperbolic and a nonhyperbolic chaotic attractor. The statistical thermodynamics of local Lyapunov exponents is also studied and a relation between the thermodynamics of scaling indices and of local Lyapunov exponents is invetigated. For the nonhyperbolic example, the free energy and entropy functions of local Lyapunov exponents are obtained in analytic forms. These results display the existence of phase transitions. A phase transition is seen in the thermodynamics of scaling indices also.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37A60 | Dynamical aspects of statistical mechanics |
| 82B30 | Statistical thermodynamics |
Keywords:
chaos; natural measure; scaling index; symbol sequence; one-dimensional lattice system; thermodynamic approach; generalized dimension; local Lyapunov exponent; generalized entropy; nonhyperbolic attractor; phase transition; scaling lawReferences:
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