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

Abstract

Let $Φ$ be a quasi-periodically forced quadratic map, where the rotation constant $ω$ is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under $Φ$) attracting graph of a nowhere continuous measurable function $ψ$ from the circle $\mathbb{T}$ to $[0,1]$. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter $β$ approaches a critical value $β_0$, 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 $β$, as $β$ approaches the critical parameter value $β_0$ 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.