Dynamics towards the Feigenbaum attractor

TL;DR

This paper provides a detailed analysis of the dynamics leading to the Feigenbaum attractor, revealing complex preimage structures, fractal boundaries, and power-law behaviors in the convergence process.

Contribution

It uncovers the intricate preimage and gap structures associated with the Feigenbaum attractor, offering new insights into the hierarchical and fractal nature of the dynamics.

Findings

Preimages of attractor and repellor are densely embedded.

Preimage layout forms from supercycle attractor boundaries.

Gaps in phase space follow power-law distributions with log-periodic modulation.

Abstract

We expose at a previously unknown level of detail the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated.