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On the reducibility of a class of almost periodic Hamiltonian systems
a. | School of Mathematics and Statistics, Xuzhou Institute of Technology, Xuzhou, 221111, China |
b. | College of Mathematics, Southeast University, Nanjing, 210096, China |
$ \dot{x} = (A+\varepsilon Q(t, \varepsilon))x, \; x\in R^{2}, $ |
$ A $ |
$ Q(t, \varepsilon) $ |
$ t $ |
$ \varepsilon $ |
$ \varepsilon $ |
References:
[1] |
N. N. Bogoljubov, J. A. Mitropoliski and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976. |
[2] |
L. H. Eliasson, Floquent solutions for the one-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447–482.
doi: 10.1007/BF02097013. |
[3] |
H.-L. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dyn. Differ. Equ., 20 (2008), 831–866.
doi: 10.1007/s10884-008-9113-6. |
[4] |
R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperodic linear differential systems, J. Dyn. Differ. Equ., 41 (1981), 262–288.
doi: 10.1016/0022-0396(81)90062-0. |
[5] |
A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, 98 (1992), 111–124.
doi: 10.1016/0022-0396(92)90107-X. |
[6] |
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. |
[7] |
J. Li and C. Zhu, On the reducibility of a class of finitely differentiable quasi-periodic linear systems, J. Math. Anal. Appl., 413 (2014), 69–83.
doi: 10.1016/j.jmaa.2013.10.077. |
[8] |
J. Li, C. Zhu and S. Chen, On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium, Qual. Theory Dyn. Syst., 16 (2017), 127–147.
doi: 10.1007/s12346-015-0164-x. |
[9] |
J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393.
doi: 10.1007/BF02096763. |
[10] |
H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theor. Dyn. Syst., 24 (2004), 1787–1832.
doi: 10.1017/S0143385703000774. |
[11] |
X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318–2329.
doi: 10.1016/j.na.2007.08.016. |
[12] |
H. Whitney, Analytical extensions of differentiable functions defined in closed sets, Trans. A. M. S., 36 (1934), 63–89.
doi: 10.1090/S0002-9947-1934-1501735-3. |
[13] |
J. Xu and J. You, On reducibility of linear differential equations with almost-periodic coefficients, Chinese Ann. Math. A(in Chinese), 17 (1996), 607–616. |
[14] |
J. Xu, On the reducibility of a class of linear differential equations with quasiperiodic coefficients, Mathematika, 46 (1999), 443–451.
doi: 10.1112/S0025579300007907. |
[15] |
J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theor. Dyn. Syst., 35 (2015), 2334–2352.
doi: 10.1017/etds.2014.31. |
[16] |
J. Xu, K. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, P. Am. Math. Soc., 144 (2016), 4793–4805.
doi: 10.1090/proc/13088. |
[17] |
L. Zhang and J. Xu, Persistence of invariant tori in Hamiltonian systems with two-degree of freedom, J. Math. Anal. Appl., 338 (2008), 793-802.
doi: 10.1016/j.jmaa.2007.05.052. |
show all references
References:
[1] |
N. N. Bogoljubov, J. A. Mitropoliski and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976. |
[2] |
L. H. Eliasson, Floquent solutions for the one-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447–482.
doi: 10.1007/BF02097013. |
[3] |
H.-L. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dyn. Differ. Equ., 20 (2008), 831–866.
doi: 10.1007/s10884-008-9113-6. |
[4] |
R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperodic linear differential systems, J. Dyn. Differ. Equ., 41 (1981), 262–288.
doi: 10.1016/0022-0396(81)90062-0. |
[5] |
A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, 98 (1992), 111–124.
doi: 10.1016/0022-0396(92)90107-X. |
[6] |
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. |
[7] |
J. Li and C. Zhu, On the reducibility of a class of finitely differentiable quasi-periodic linear systems, J. Math. Anal. Appl., 413 (2014), 69–83.
doi: 10.1016/j.jmaa.2013.10.077. |
[8] |
J. Li, C. Zhu and S. Chen, On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium, Qual. Theory Dyn. Syst., 16 (2017), 127–147.
doi: 10.1007/s12346-015-0164-x. |
[9] |
J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393.
doi: 10.1007/BF02096763. |
[10] |
H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theor. Dyn. Syst., 24 (2004), 1787–1832.
doi: 10.1017/S0143385703000774. |
[11] |
X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318–2329.
doi: 10.1016/j.na.2007.08.016. |
[12] |
H. Whitney, Analytical extensions of differentiable functions defined in closed sets, Trans. A. M. S., 36 (1934), 63–89.
doi: 10.1090/S0002-9947-1934-1501735-3. |
[13] |
J. Xu and J. You, On reducibility of linear differential equations with almost-periodic coefficients, Chinese Ann. Math. A(in Chinese), 17 (1996), 607–616. |
[14] |
J. Xu, On the reducibility of a class of linear differential equations with quasiperiodic coefficients, Mathematika, 46 (1999), 443–451.
doi: 10.1112/S0025579300007907. |
[15] |
J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theor. Dyn. Syst., 35 (2015), 2334–2352.
doi: 10.1017/etds.2014.31. |
[16] |
J. Xu, K. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, P. Am. Math. Soc., 144 (2016), 4793–4805.
doi: 10.1090/proc/13088. |
[17] |
L. Zhang and J. Xu, Persistence of invariant tori in Hamiltonian systems with two-degree of freedom, J. Math. Anal. Appl., 338 (2008), 793-802.
doi: 10.1016/j.jmaa.2007.05.052. |
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