# American Institute of Mathematical Sciences

November  2016, 36(11): 6599-6622. doi: 10.3934/dcds.2016086

## 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, China 2 Departament of Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona 3 School of Mathematics, Shandong University, Jinan, Shandong 250100

Received  November 2013 Revised  May 2016 Published  August 2016

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.
Citation: Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086
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##### References:
 [1] I. Baldomà, E. Fontich and P. Martín, Stable manifolds for parabolic points through the parameterization method,, Preprint., (). Google Scholar [2] M. Ding, C. Grebogi and E. Ott, Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange non-chaotic to chaotic,, Phys. Rev. A, 39 (1989), 2593. doi: 10.1103/PhysRevA.39.2593. Google Scholar [3] P. Glendinning, The non-smooth pitchork bifurcation,, Discrete Contin. Dyn. Syst. Ser. B, 6 (2002), 1. Google Scholar [4] M. Guardia, P. Martín and T.M. Seara, Oscillatory motions for the restricted planar circular three body problem,, Invent. Math., 203 (2016), 417. doi: 10.1007/s00222-015-0591-y. Google Scholar [5] À. Haro and J. Puig, Strange nonchaotic attractors in Harper maps,, Chaos, 16 (2006). doi: 10.1063/1.2259821. Google Scholar [6] À. Jorba, P. Rabassa and J. C. Tatjer, Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 589. doi: 10.3934/dcds.2014.34.589. Google Scholar [7] À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients,, J. Differential Equations, 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar [8] À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points,, SIAM J. Math. Anal., 27 (1996), 1704. doi: 10.1137/S0036141094276913. Google Scholar [9] À. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537. doi: 10.3934/dcdsb.2008.10.537. Google Scholar [10] À. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations,, J. Nonlinear Sci., 7 (1997), 427. doi: 10.1007/s003329900036. Google Scholar [11] L. M. Lerman, On remarks of skew products over irrational rotation,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3675. doi: 10.1142/S0218127405014118. Google Scholar [12] R. Martínez and C. Pinyol, Parabolic orbits in the elliptic restricted three body problem,, J. Differential Equations, 111 (1994), 299. doi: 10.1006/jdeq.1994.1084. Google Scholar [13] R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion,, Regul. Chaotic Dyn., 19 (2014), 745. doi: 10.1134/S1560354714060112. Google Scholar [14] U. Vaidya and I. Mezić, Existence of invariant tori in three dimensional maps with degeneracy,, Phys. D, 241 (2012), 1136. doi: 10.1016/j.physd.2012.03.004. Google Scholar [15] J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point,, J. Differential Equations, 250 (2011), 551. doi: 10.1016/j.jde.2010.09.030. Google Scholar [16] J. Xu, On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar system,, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2593. doi: 10.3934/dcds.2013.33.2593. Google Scholar [17] J. Xu and S. Jiang, Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation,, Ergodic Theory Dynam. Systems, 31 (2011), 599. doi: 10.1017/S0143385709001114. Google Scholar [18] J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate,, Proc. Amer. Math. Soc., 126 (1998), 1445. doi: 10.1090/S0002-9939-98-04523-7. Google Scholar [19] J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in hamiltonian systems,, Comm. Math. Phys., 192 (1998), 145. doi: 10.1007/s002200050294. Google Scholar
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