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Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems
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 |
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. |
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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