Labyrinthine pathways towards supercycle attractors in unimodal maps

TL;DR

This paper explores the complex fractal structures and hierarchical properties of superstable cycles in unimodal maps, revealing new insights into their basins of attraction, renormalization fixed points, and ultra-fast convergence dynamics.

Contribution

It uncovers novel fractal and hierarchical properties of superstable cycles, introduces a closed-form $q$-exponential fixed-point map, and characterizes ultra-fast convergence dynamics.

Findings

Basins of attraction develop fractal boundaries with increasing complexity.

Hierarchical structures in fractal boundaries exhibit exponential clustering.

Ultra-fast convergence to attractors involves exponential of exponential decay.

Abstract

We uncover previously unknown properties of the family of periodic superstable cycles in unimodal maps characterized each by a Lyapunov exponent that diverges to minus infinity. Amongst the main novel properties are the following: i) The basins of attraction for the phases of the cycles develop fractal boundaries of increasing complexity as the period-doubling structure advances towards the transition to chaos. ii) The fractal boundaries, formed by the preimages of the repellor, display hierarchical structures organized according to exponential clusterings that manifest in the dynamics as sensitivity to the final state and transient chaos. iii) There is a functional composition renormalization group (RG) fixed-point map associated to the family of supercycles. iv) This map is given in closed form by the same kind of $q$-exponential function found for both the pitchfork and tangent bifurcation attractors. v) There is a final stage ultra-fast dynamics towards the attractor with a sensitivity to initial conditions that decreases as an exponential of an exponential of time.