-
Previous Article
Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems
- DCDS-B Home
- This Issue
-
Next Article
A diffusive weak Allee effect model with U-shaped emigration and matrix hostility
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. |
[1] |
John Leventides, Costas Poulios, Georgios Alkis Tsiatsios, Maria Livada, Stavros Tsipras, Konstantinos Lefcaditis, Panagiota Sargenti, Aleka Sargenti. Systems theory and analysis of the implementation of non pharmaceutical policies for the mitigation of the COVID-19 pandemic. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021004 |
[2] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[3] |
Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 |
[4] |
Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 |
[5] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[6] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[7] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
[8] |
Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029 |
[9] |
W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 |
[10] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[11] |
Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051 |
[12] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[13] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[14] |
Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
[15] |
V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153 |
[16] |
Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 |
[17] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[18] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[19] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[20] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]