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
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

show all references

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

[1]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177

[2]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683

[3]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233

[4]

Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043

[5]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038

[6]

Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure & Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027

[7]

Inmaculada Baldomá, Ernest Fontich, Rafael de la Llave, Pau Martín. The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 835-865. doi: 10.3934/dcds.2007.17.835

[8]

Paula Kemp. Fixed points and complete lattices. Conference Publications, 2007, 2007 (Special) : 568-572. doi: 10.3934/proc.2007.2007.568

[9]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443

[10]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371

[11]

Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287

[12]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761

[13]

M. Burak Erdoğan, Nikolaos Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2669-2684. doi: 10.3934/cpaa.2013.12.2669

[14]

M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343

[15]

Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517

[16]

Gemma Huguet, Rafael de la Llave, Yannick Sire. Fast iteration of cocycles over rotations and computation of hyperbolic bundles. Conference Publications, 2013, 2013 (special) : 323-333. doi: 10.3934/proc.2013.2013.323

[17]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911

[18]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683

[19]

Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593

[20]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]