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A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces
Response solutions for degenerate reversible harmonic oscillators
School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
$ \ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R}, $ |
$ \lambda = \pm 1 $ |
$ n>1 $ |
$ f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon) $ |
$ f $ |
$ \lambda = 1 $ |
$ \epsilon $ |
$ \omega $ |
$ \lambda = -1 $ |
$ \epsilon_* $ |
$ \mathcal{E}\in(0, \epsilon_*) $ |
$ \epsilon\in\mathcal{E} $ |
$ \omega $ |
$ f $ |
References:
[1] |
B. L. J. Braaksma and H. W. Broer,
On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 115-168.
doi: 10.1016/S0294-1449(16)30370-5. |
[2] |
H. W. Broer, M. C. Ciocci and H. Hanßmann,
The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623.
doi: 10.1142/S021812740701866X. |
[3] |
L. Corsi and G. Gentile,
Oscillator synchronisation under arbitrary quasi-periodic forcing, Comm. Math. Phys., 316 (2012), 489-529.
doi: 10.1007/s00220-012-1548-2. |
[4] |
L. Corsi and G. Gentile, Resonant tori of arbitrary codimension for quasi-periodically forced systems, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 3, 21 pp.
doi: 10.1007/s00030-016-0425-7. |
[5] |
M. Friedman,
Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Amer. Math. Soc., 73 (1967), 460-464.
doi: 10.1090/S0002-9904-1967-11783-X. |
[6] |
G. Gentile,
Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.
doi: 10.1017/S0143385706000757. |
[7] |
G. Gentile,
Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.
doi: 10.1017/S0143385709000583. |
[8] |
G. Gentile,
Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808.
|
[9] |
Y. Han, Y. Li and Y. Yi,
Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations, 227 (2006), 670-691.
doi: 10.1016/j.jde.2006.02.006. |
[10] |
H. Hanßmann,
Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.
doi: 10.1134/S1560354710520059. |
[11] |
S. Hu and B. Liu,
Degenerate lower dimensional invariant tori in reversible system, Discrete Contin. Dyn. Syst., 38 (2018), 3735-3763.
doi: 10.3934/dcds.2018162. |
[12] |
S. Hu and B. Liu,
Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J. Differential Equations, 266 (2019), 7459-7480.
doi: 10.1016/j.jde.2018.12.001. |
[13] |
Z. Lou and J. Geng,
Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differential Equations, 263 (2017), 3894-3927.
doi: 10.1016/j.jde.2017.05.007. |
[14] |
J. Moser,
Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[15] |
J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950. |
[16] |
W. Si and J. Si,
Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations, J. Differential Equations, 262 (2017), 4771-4822.
doi: 10.1016/j.jde.2016.12.019. |
[17] |
W. Si and Y. Yi,
Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.
doi: 10.1088/1361-6544/aba093. |
[18] |
J. Wang, J. You and Q. Zhou,
Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.
doi: 10.1090/tran/6800. |
[19] |
J. You,
A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192 (1998), 145-168.
doi: 10.1007/s002200050294. |
[20] |
X. Wang, J. Xu and D. Zhang,
Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012), 776-790.
doi: 10.1016/j.jmaa.2011.09.030. |
[21] |
X. Wang, J. Xu and D. Zhang,
On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.
doi: 10.1017/etds.2014.34. |
show all references
References:
[1] |
B. L. J. Braaksma and H. W. Broer,
On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 115-168.
doi: 10.1016/S0294-1449(16)30370-5. |
[2] |
H. W. Broer, M. C. Ciocci and H. Hanßmann,
The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623.
doi: 10.1142/S021812740701866X. |
[3] |
L. Corsi and G. Gentile,
Oscillator synchronisation under arbitrary quasi-periodic forcing, Comm. Math. Phys., 316 (2012), 489-529.
doi: 10.1007/s00220-012-1548-2. |
[4] |
L. Corsi and G. Gentile, Resonant tori of arbitrary codimension for quasi-periodically forced systems, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Paper No. 3, 21 pp.
doi: 10.1007/s00030-016-0425-7. |
[5] |
M. Friedman,
Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Amer. Math. Soc., 73 (1967), 460-464.
doi: 10.1090/S0002-9904-1967-11783-X. |
[6] |
G. Gentile,
Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.
doi: 10.1017/S0143385706000757. |
[7] |
G. Gentile,
Quasi-periodic motions in strongly dissipative forced systems, Ergodic Theory Dynam. Systems, 30 (2010), 1457-1469.
doi: 10.1017/S0143385709000583. |
[8] |
G. Gentile,
Construction of quasi-periodic response solutions in forced strongly dissipative systems, Forum Math., 24 (2012), 791-808.
|
[9] |
Y. Han, Y. Li and Y. Yi,
Degenerate lower-dimensional tori in Hamiltonian systems, J. Differential Equations, 227 (2006), 670-691.
doi: 10.1016/j.jde.2006.02.006. |
[10] |
H. Hanßmann,
Quasi-periodic bifurcations in reversible systems, Regul. Chaotic Dyn., 16 (2011), 51-60.
doi: 10.1134/S1560354710520059. |
[11] |
S. Hu and B. Liu,
Degenerate lower dimensional invariant tori in reversible system, Discrete Contin. Dyn. Syst., 38 (2018), 3735-3763.
doi: 10.3934/dcds.2018162. |
[12] |
S. Hu and B. Liu,
Completely degenerate lower-dimensional invariant tori for Hamiltonian system, J. Differential Equations, 266 (2019), 7459-7480.
doi: 10.1016/j.jde.2018.12.001. |
[13] |
Z. Lou and J. Geng,
Quasi-periodic response solutions in forced reversible systems with Liouvillean frequencies, J. Differential Equations, 263 (2017), 3894-3927.
doi: 10.1016/j.jde.2017.05.007. |
[14] |
J. Moser,
Combination tones for Duffings equation, Comm. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[15] |
J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publisher, New York, 1950. |
[16] |
W. Si and J. Si,
Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations, J. Differential Equations, 262 (2017), 4771-4822.
doi: 10.1016/j.jde.2016.12.019. |
[17] |
W. Si and Y. Yi,
Completely degenerate responsive tori in Hamiltonian systems, Nonlinearity, 33 (2020), 6072-6098.
doi: 10.1088/1361-6544/aba093. |
[18] |
J. Wang, J. You and Q. Zhou,
Response solutions for quasi-periodically forced harmonic oscillators, Trans. Amer. Math. Soc., 369 (2017), 4251-4274.
doi: 10.1090/tran/6800. |
[19] |
J. You,
A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192 (1998), 145-168.
doi: 10.1007/s002200050294. |
[20] |
X. Wang, J. Xu and D. Zhang,
Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012), 776-790.
doi: 10.1016/j.jmaa.2011.09.030. |
[21] |
X. Wang, J. Xu and D. Zhang,
On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.
doi: 10.1017/etds.2014.34. |
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