Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family

TL;DR

This paper studies the transition from smooth to strange attractors in a quasi-periodically forced quadratic map, revealing power law asymptotics in the bifurcation process and the behavior of invariant curves.

Contribution

It provides a detailed analysis of the asymptotic behavior of invariant curves during the bifurcation from smooth to strange attractors in a quasi-periodic setting.

Findings

Minimum distance between colliding invariant curves decreases linearly as parameter approaches critical value.

Growth of the derivative of the attracting graph is bounded by the reciprocal of the square root of the minimum distance.

Results elucidate the power law asymptotics in the creation of strange attractors.

Abstract

Let $\Phi$ be a quasi-periodically forced quadratic map, where the rotation constant $\omega$ is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under $\Phi$) attracting graph of a nowhere continuous measurable function $\psi$ from the circle $\mathbb{T}$ to $[0,1]$. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter $\beta$ approaches a critical value $\beta_0$, and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller. Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter $\beta$, as $\beta$ approaches the critical parameter value $\beta_0$ from below. Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.