Power law asymptotics in the creation of strange attractors in the quasiperiodically forced quadratic family
Abstract
Let $\Phi$ be a quasiperiodically forced quadratic map, where the rotation constant $\omega$ is a Diophantine irrational. A strange nonchaotic 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 socalled 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 DOI:
 10.48550/arXiv.1503.05822
 arXiv:
 arXiv:1503.05822
 Bibcode:
 2015arXiv150305822O
 Keywords:

 Mathematics  Dynamical Systems