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

¹ Instituto de Física, Universidad Nacional Autónoma de México - Apartado Postal 20-364, México 01000 D.F., Mexico
² Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México - Apartado Postal 20-364, México 01000 D.F., Mexico

Received: 14 November 2013
Accepted: 11 February 2014

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 , periods can be expressed as power laws or deformed exponentials. For a finite period the structure is undeveloped in the sense that there is no true configurational degeneracy, but in the limit 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.