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

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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

Tingting Zhang ^1, , Àngel Jorba ^2, and Jianguo Si ^3,

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

Abstract
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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.

Keywords: weakly hyperbolic invariant torus, degenerate fixed points, Quasiperiodic systems, KAM iteration..

Mathematics Subject Classification: Primary: 34K14, 34K19; Secondary: 34K2.

Citation: Tingting Zhang, Àngel Jorba, Jianguo Si. Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6599-6622. doi: 10.3934/dcds.2016086

References:

[1]	I. Baldomà, E. Fontich and P. Martín, Stable manifolds for parabolic points through the parameterization method, Preprint.
[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), 2593-2598. doi: 10.1103/PhysRevA.39.2593.
[3]	P. Glendinning, The non-smooth pitchork bifurcation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2002), 1-7.
[4]	M. Guardia, P. Martín and T.M. Seara, Oscillatory motions for the restricted planar circular three body problem, Invent. Math., 203 (2016), 417-492. doi: 10.1007/s00222-015-0591-y.
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[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), 589-597. doi: 10.3934/dcds.2014.34.589.
[7]	À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.
[8]	À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737. doi: 10.1137/S0036141094276913.
[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), 537-567. doi: 10.3934/dcdsb.2008.10.537.
[10]	À. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci., 7 (1997), 427-473. doi: 10.1007/s003329900036.
[11]	L. M. Lerman, On remarks of skew products over irrational rotation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3675-3689. doi: 10.1142/S0218127405014118.
[12]	R. Martínez and C. Pinyol, Parabolic orbits in the elliptic restricted three body problem, J. Differential Equations, 111 (1994), 299-339. doi: 10.1006/jdeq.1994.1084.
[13]	R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion, Regul. Chaotic Dyn., 19 (2014), 745-765. doi: 10.1134/S1560354714060112.
[14]	U. Vaidya and I. Mezić, Existence of invariant tori in three dimensional maps with degeneracy, Phys. D, 241 (2012), 1136-1145. doi: 10.1016/j.physd.2012.03.004.
[15]	J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2011), 551-571. doi: 10.1016/j.jde.2010.09.030.
[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), 2593-2619. doi: 10.3934/dcds.2013.33.2593.
[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), 599-611. doi: 10.1017/S0143385709001114.
[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), 1445-1451. doi: 10.1090/S0002-9939-98-04523-7.
[19]	J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in hamiltonian systems, Comm. Math. Phys., 192 (1998), 145-168. doi: 10.1007/s002200050294.

show all references

References:

[1]	I. Baldomà, E. Fontich and P. Martín, Stable manifolds for parabolic points through the parameterization method, Preprint.
[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), 2593-2598. doi: 10.1103/PhysRevA.39.2593.
[3]	P. Glendinning, The non-smooth pitchork bifurcation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2002), 1-7.
[4]	M. Guardia, P. Martín and T.M. Seara, Oscillatory motions for the restricted planar circular three body problem, Invent. Math., 203 (2016), 417-492. doi: 10.1007/s00222-015-0591-y.
[5]	À. Haro and J. Puig, Strange nonchaotic attractors in Harper maps, Chaos, 16 (2006). Available from: https://www.researchgate.net/publication/6781348_Strange_Nonchaotic_Attractors_in_Harper_Maps. doi: 10.1063/1.2259821.
[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), 589-597. doi: 10.3934/dcds.2014.34.589.
[7]	À. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X.
[8]	À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737. doi: 10.1137/S0036141094276913.
[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), 537-567. doi: 10.3934/dcdsb.2008.10.537.
[10]	À. Jorba and J. Villanueva, On the persistence of lower dimensional invariant tori under quasi-periodic perturbations, J. Nonlinear Sci., 7 (1997), 427-473. doi: 10.1007/s003329900036.
[11]	L. M. Lerman, On remarks of skew products over irrational rotation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3675-3689. doi: 10.1142/S0218127405014118.
[12]	R. Martínez and C. Pinyol, Parabolic orbits in the elliptic restricted three body problem, J. Differential Equations, 111 (1994), 299-339. doi: 10.1006/jdeq.1994.1084.
[13]	R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion, Regul. Chaotic Dyn., 19 (2014), 745-765. doi: 10.1134/S1560354714060112.
[14]	U. Vaidya and I. Mezić, Existence of invariant tori in three dimensional maps with degeneracy, Phys. D, 241 (2012), 1136-1145. doi: 10.1016/j.physd.2012.03.004.
[15]	J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2011), 551-571. doi: 10.1016/j.jde.2010.09.030.
[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), 2593-2619. doi: 10.3934/dcds.2013.33.2593.
[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), 599-611. doi: 10.1017/S0143385709001114.
[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), 1445-1451. doi: 10.1090/S0002-9939-98-04523-7.
[19]	J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in hamiltonian systems, Comm. Math. Phys., 192 (1998), 145-168. doi: 10.1007/s002200050294.

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