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]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, , () : -. doi: 10.3934/era.2021001

[2]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[3]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226

[4]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[5]

Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329

[6]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561

[7]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[8]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[9]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[10]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139

[11]

Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270

[12]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[13]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[14]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030

[15]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[16]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409

[17]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[19]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[20]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]