Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos

Abstract

We consider both the dynamics within and towards the supercycle attractors along the perioddoubling route to chaos to analyze the development of a statistical-mechanical structure. In this structure the partition function consists of the sum of the attractor position distances known as supercycle diameters and the associated thermodynamic potential measures the rate of approach of trajectories to the attractor. The configurational weights for finite 2 N , and infinite N → ∞, periods can be expressed as power laws or deformed exponentials. For finite period the structure is undeveloped in the sense that there is no true configurational degeneracy, but in the limit N → ∞ this is realized together with the analog property of a Legendre transform linking entropies of two ensembles. We also study the partition functions for all N and the action of the Central Limit Theorem via a binomial approximation. PACS 5.45.Ac Low-dimensional chaos PACS 05.20.Gg Classical ensemble theory PACS 05.45.Df Fractals

A zodiac of studies on complex systems

“…From the start, the property that summarized our calculations for the dynamics towards the periodic attractors of the quadratic map resembled in form that of a partition function. Continued work supported this interpretation and finally a model for selforganization materialized [47][48][49][50][51][52][53][54]. Diagonals.…”

“…Only recently [53,54], the first stage in the statisticalmechanical justification of Eq. (30) as a bona fide partition function was put together.…”

Conduction at the onset of chaos

“…Clarification of the latter result demands for a better construction of the invariant (equilibrium) measure than the one obtained by simply taking the long-time limit of the particles' distribution. More generally, a statistical mechanics approach [22] linking the microscopic dynamics (e.g., in terms of the Lyapunov or generalized Lyapunov exponents) to the observed nonequilibrium thermodynamics is an intriguing open question.…”

Pascal (Yang Hui) triangles and power laws in the logistic map

“…The diameters naturally assemble into well-defined size groups and the numbers of them in each group can be precisely arranged into a Pascal Triangle. In fact, the diameter lengths within each group are not equal, however the differences in lengths within groups diminishes rapidly as the period 2 n increases [10]. A detailed study of the quantitative differences between the values of the diameters generated by the logistic map and those obtained from the binomial approximation that form the Pascal triangle in Fig.…”

“…A detailed study of the quantitative differences between the values of the diameters generated by the logistic map and those obtained from the binomial approximation that form the Pascal triangle in Fig. 1 is given in [10]. There are two groups with only one member, the largest and the shortest diameters, and the numbers within each group are given by the binomial coefficients.…”