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January  2020, 19(1): 541-585. doi: 10.3934/cpaa.2020027

Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method

Wen Si , Fenfen Wang and  Jianguo Si ,

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

* Corresponding author

Received  July 2018 Revised  January 2019 Published  July 2019

Fund Project: The first author was supported by a CSC scholarship (CSC student ID: 201606240030). The third author was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171185, 11571201).

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 [14,13,23,20] from quasi-periodic case to almost-periodic case and also give rise to the reducibility of almost-periodic perturbed linear differential systems.

Keywords: Almost-periodic response solutions, Pöschel-Rüssmann KAM method, Herman method, weaker non-resonant condition.
Mathematics Subject Classification: Primary: 34J40, 34C27; Secondary: 34E20.
Citation: Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure & Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, 2nd Edition, Robert E. Krieger Publishing Co., Huntington, NY, 1980.  Google Scholar

[7]

H. Hanßmann, The quasi-periodic centre-saddle bifurcation, J. Differential Equations, 142 (1998), 305-370.  doi: 10.1006/jdeq.1997.3365.  Google Scholar

[8]

H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems, Lecture Notes in Matematics, vol. 1893, Springer, Heidelberg, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Pöchel, Small divisors with spatial structure in infinite dimensional dynamical Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-395.   Google Scholar

[16]

J. Pöschel, KAM á la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. K. Hale, Ordinary Differential Equations, 2nd Edition, Robert E. Krieger Publishing Co., Huntington, NY, 1980.  Google Scholar

[7]

H. Hanßmann, The quasi-periodic centre-saddle bifurcation, J. Differential Equations, 142 (1998), 305-370.  doi: 10.1006/jdeq.1997.3365.  Google Scholar

[8]

H. Hanßmann, Local and Semi-local Bifurcations in Hamiltonian Dynamical Systems, Lecture Notes in Matematics, vol. 1893, Springer, Heidelberg, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Pöchel, Small divisors with spatial structure in infinite dimensional dynamical Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-395.   Google Scholar

[16]

J. Pöschel, KAM á la R, Regul. Chaotic Dyn., 16 (2011), 17-23.  doi: 10.1134/S1560354710520060.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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