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Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian
Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method
School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
This paper focuses on almost-periodic time-dependent perturbations of a class of almost-periodically forced systems near non-hyperbolic equilibrium points in two cases: (a) elliptic case, (b) degenerate case (including completely degenerate). In elliptic case, it is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to perturbation parameter $ \epsilon, $ there exists a Cantor set $ \mathcal{E}\subset (0, \epsilon_0) $ of positive Lebesgue measure with sufficiently small $ \epsilon_0 $ such that for each $ \epsilon\in\mathcal{E} $ the system has an almost-periodic response solution. In degenerate case, we prove that, firstly, the almost-periodically perturbed degenerate system in one-dimensional case admits an almost-periodic response solution under nonzero average condition on perturbation and some weak non-resonant condition; Secondly, imposing further restriction on smallness of the perturbation besides nonzero average, we prove the almost-periodically forced degenerate system in $ n $-dimensional case has an almost-periodic response solution under small perturbation without any non-resonant condition; Finally, almost-periodic response solution can still be obtained with weakened nonzero average condition by used Herman method but non-resonant condition should be strengthened. Some proofs of main results are based on a modified Pöschel-Rüssmann KAM method, our results show that Pöschel-Rüssmann KAM method can be applied to study the existence of almost-periodic solutions for almost-periodically forced non-conservative systems. Our results generalize the works in [
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Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc., 83 (1990), 1-175.
doi: 10.1090/memo/0421. |
[2] |
H. W. Broer, H. Hanßmann and J. You,
Bifurcations of normally parabolic in Hamiltonian systems, Nonlinearity, 18 (2005), 1735-1769.
doi: 10.1088/0951-7715/18/4/018. |
[3] |
H. W. Broer, H. Hanßmann, A. Jorba, J. Villanueva and F. Wagener,
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity, 16 (2003), 1751-1791.
doi: 10.1088/0951-7715/16/5/312. |
[4] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems, Lecture Notes in Matematics, vol. 1645, Springer-Heidelberg, 1645. |
[5] |
H. W. Broer, H. Hanßmann and J. You,
Umbilical torus bifurcations in Hamiltonian systems, J. Differential Equations, 222 (2006), 233-262.
doi: 10.1016/j.jde.2005.06.030. |
[6] |
J. K. Hale, Ordinary Differential Equations, 2nd Edition, Robert E. Krieger Publishing Co., Huntington, NY, 1980. |
[7] |
H. Hanßmann,
The quasi-periodic centre-saddle bifurcation, J. Differential Equations, 142 (1998), 305-370.
doi: 10.1006/jdeq.1997.3365. |
[8] |
H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems, Lecture Notes in Matematics, vol. 1893, Springer, Heidelberg, 2007. |
[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] |
P. Huang, X. Li and B. Liu,
Almost periodic solutions for an asymmetric oscillation, J. Differential Equations, 263 (2017), 8916-8946.
doi: 10.1016/j.jde.2017.08.063. |
[11] |
P. Huang, X. Li and B. Liu,
Invariant curves of smooth quasi-periodic mappings, Discrete Continuous Dynam. Systems - A, 38 (2018), 131-154.
doi: 10.3934/dcds.2018006. |
[12] |
R. A. Johnson and G. R. Sell,
Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differential Equations, 41 (1981), 262-288.
doi: 10.1016/0022-0396(81)90062-0. |
[13] |
A. 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. |
[14] |
A. Jorba and C. Simó,
On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.
doi: 10.1137/S0036141094276913. |
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J. Pöchel,
Small divisors with spatial structure in infinite dimensional dynamical Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-395.
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[17] |
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KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718.
doi: 10.3934/dcdss.2010.3.683. |
[18] |
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. |
[19] |
W. Si and J. Si,
Elliptic-type degenerate invariant tori for quasi-periodically forced four-dimensional non-conservative systems, J. Math. Anal. Appl., 460 (2018), 164-202.
doi: 10.1016/j.jmaa.2017.11.047. |
[20] |
W. Si and J. Si,
Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.
doi: 10.1088/1361-6544/aaa7b9. |
[21] |
F. Wagener,
On the quasi-periodic d-fold degenerate bifurcation, J. Differential Equations, 216 (2005), 216-281.
doi: 10.1016/j.jde.2005.06.013. |
[22] |
J. Xu and J. You,
Reducibility of linear differential equations with almost periodic coefficients, (Chinese) Chinese Ann. Math. Ser. A, 17 (1996), 607-616.
|
[23] |
J. Xu and S. Jiang,
Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergod. Th. & Dynam. Sys., 31 (2010), 599-611.
doi: 10.1017/S0143385709001114. |
[24] |
J. Xu,
On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2010), 551-571.
doi: 10.1016/j.jde.2010.09.030. |
[25] |
J. Xu,
On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems, Discrete Continuous Dynam. Systems - A, 33 (2013), 2593-2619.
doi: 10.3934/dcds.2013.33.2593. |
[26] |
J. Xu, J. You and Q. Qiu,
Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-386.
doi: 10.1007/PL00004344. |
[27] |
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. |
[28] |
T. Zhang, A. Jorba and J. Si,
Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case, Discrete Continuous Dynam. Systems - A, 36 (2016), 6599-6622.
doi: 10.3934/dcds.2016086. |
show all references
References:
[1] |
H. W. Broer, G. B. Huitema, F. Takens and B. J. L. Braaksma,
Unfoldings and bifurcations of quasi-periodic tori, Mem. Amer. Math. Soc., 83 (1990), 1-175.
doi: 10.1090/memo/0421. |
[2] |
H. W. Broer, H. Hanßmann and J. You,
Bifurcations of normally parabolic in Hamiltonian systems, Nonlinearity, 18 (2005), 1735-1769.
doi: 10.1088/0951-7715/18/4/018. |
[3] |
H. W. Broer, H. Hanßmann, A. Jorba, J. Villanueva and F. Wagener,
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity, 16 (2003), 1751-1791.
doi: 10.1088/0951-7715/16/5/312. |
[4] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems, Lecture Notes in Matematics, vol. 1645, Springer-Heidelberg, 1645. |
[5] |
H. W. Broer, H. Hanßmann and J. You,
Umbilical torus bifurcations in Hamiltonian systems, J. Differential Equations, 222 (2006), 233-262.
doi: 10.1016/j.jde.2005.06.030. |
[6] |
J. K. Hale, Ordinary Differential Equations, 2nd Edition, Robert E. Krieger Publishing Co., Huntington, NY, 1980. |
[7] |
H. Hanßmann,
The quasi-periodic centre-saddle bifurcation, J. Differential Equations, 142 (1998), 305-370.
doi: 10.1006/jdeq.1997.3365. |
[8] |
H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems, Lecture Notes in Matematics, vol. 1893, Springer, Heidelberg, 2007. |
[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] |
P. Huang, X. Li and B. Liu,
Almost periodic solutions for an asymmetric oscillation, J. Differential Equations, 263 (2017), 8916-8946.
doi: 10.1016/j.jde.2017.08.063. |
[11] |
P. Huang, X. Li and B. Liu,
Invariant curves of smooth quasi-periodic mappings, Discrete Continuous Dynam. Systems - A, 38 (2018), 131-154.
doi: 10.3934/dcds.2018006. |
[12] |
R. A. Johnson and G. R. Sell,
Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems, J. Differential Equations, 41 (1981), 262-288.
doi: 10.1016/0022-0396(81)90062-0. |
[13] |
A. 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. |
[14] |
A. Jorba and C. Simó,
On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.
doi: 10.1137/S0036141094276913. |
[15] |
J. Pöchel,
Small divisors with spatial structure in infinite dimensional dynamical Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-395.
|
[16] |
J. Pöschel,
KAM á la R, Regul. Chaotic Dyn., 16 (2011), 17-23.
doi: 10.1134/S1560354710520060. |
[17] |
H. Rüssmann,
KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718.
doi: 10.3934/dcdss.2010.3.683. |
[18] |
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. |
[19] |
W. Si and J. Si,
Elliptic-type degenerate invariant tori for quasi-periodically forced four-dimensional non-conservative systems, J. Math. Anal. Appl., 460 (2018), 164-202.
doi: 10.1016/j.jmaa.2017.11.047. |
[20] |
W. Si and J. Si,
Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.
doi: 10.1088/1361-6544/aaa7b9. |
[21] |
F. Wagener,
On the quasi-periodic d-fold degenerate bifurcation, J. Differential Equations, 216 (2005), 216-281.
doi: 10.1016/j.jde.2005.06.013. |
[22] |
J. Xu and J. You,
Reducibility of linear differential equations with almost periodic coefficients, (Chinese) Chinese Ann. Math. Ser. A, 17 (1996), 607-616.
|
[23] |
J. Xu and S. Jiang,
Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergod. Th. & Dynam. Sys., 31 (2010), 599-611.
doi: 10.1017/S0143385709001114. |
[24] |
J. Xu,
On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2010), 551-571.
doi: 10.1016/j.jde.2010.09.030. |
[25] |
J. Xu,
On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems, Discrete Continuous Dynam. Systems - A, 33 (2013), 2593-2619.
doi: 10.3934/dcds.2013.33.2593. |
[26] |
J. Xu, J. You and Q. Qiu,
Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-386.
doi: 10.1007/PL00004344. |
[27] |
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. |
[28] |
T. Zhang, A. Jorba and J. Si,
Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case, Discrete Continuous Dynam. Systems - A, 36 (2016), 6599-6622.
doi: 10.3934/dcds.2016086. |
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