Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 34K14, 34K19; Secondary: 34K27.


    \begin{equation} \\ \end{equation}
  • [1]

    I. Baldomà, E. Fontich and P. Martín, Stable manifolds for parabolic points through the parameterization method, Preprint.


    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.


    P. Glendinning, The non-smooth pitchork bifurcation, Discrete Contin. Dyn. Syst. Ser. B, 6 (2002), 1-7.


    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.


    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.


    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.


    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.


    À. Jorba and C. Simó, On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.doi: 10.1137/S0036141094276913.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(178) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint