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 period-doubling 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 $N\to \infty$, 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 $N\to \infty$ 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.

arXiv comment11 pages, 3 figures. arXiv admin note: text overlap with arXiv:1312.0716

References in corpus (5)

• Tsallis' q index and Mori's q phase transitions at edge of chaos
• q-Deformed Statistical-Mechanical Property in the Dynamics of Trajectories en route to the Feigenbaum Attractor
• Dynamics towards the Feigenbaum attractor
• Temporal scaling at Feigenbaum points and non-extensive thermodynamics
• Labyrinthine pathways towards supercycle attractors in unimodal maps

Cited by in corpus (6)

• Generalized Statistical Mechanics at the Onset of Chaos
• Entropies for severely contracted configuration space
• Scaling of the Maximum-Entropy Turbulence Energy Spectra
• Logistic map trajectory distributions: Renormalization-group, entropy and criticality at the transition to chaos
• Pascal (Yang Hui) triangles and power laws in the logistic map
• Conduction at the onset of chaos