Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced mapsin a degenerate case

Tingting  Zhang; Àngel Jorba; Jianguo Si

doi:10.3934/dcds.2016086

Article Contents

2016, Volume 36, Issue 11: 6599-6622. Doi: 10.3934/dcds.2016086

This issue Previous Article Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems Next Article

Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case

1.
School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071
2.
Departament of Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona
3.
School of Mathematics, Shandong University, Jinan, Shandong 250100

Received: October 31, 2013

Revised: April 30, 2016

Published: May 2017

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Abstract

In this work we consider a class of degenerate analytic maps of the form \begin{eqnarray*} \left\{ \begin{array}{l} \bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ \bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ \bar{\theta}=\theta+\omega, \end{array} \right. \end{eqnarray*} where $mn>1,n\geq m,$ $h_1 \ \mbox{and} \ h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \Bbb{R}^{d}$ is a vector of rationally independent frequencies. It is shown that, under a generic non-degeneracy condition on $f$, if $\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has at least one weakly hyperbolic invariant torus.

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Mathematics Subject Classification: Primary: 34K14, 34K19; Secondary: 34K27.

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