• Previous Article
    On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model
  • DCDS-B Home
  • This Issue
  • Next Article
    On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit
doi: 10.3934/dcdsb.2020268

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

Received  October 2019 Revised  July 2020 Published  September 2020

Fund Project: The authors are supported by NSFC grant 11526177 and the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 18KJB110029

In this paper we consider the following linear almost periodic hamiltonian system
$ \dot{x} = (A+\varepsilon Q(t, \varepsilon))x, \; x\in R^{2}, $
where
$ A $
is a constant matrix with different eigenvalues, and
$ Q(t, \varepsilon) $
is analytic almost periodic with respect to
$ t $
and analytic with respect to
$ \varepsilon $
. Without any non-degeneracy condition, we prove that the linear hamiltonian system is reducible for most of sufficiently small parameter
$ \varepsilon $
by an almost periodic symplectic mapping.
Citation: Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020268
References:
[1]

N. N. Bogoljubov, J. A. Mitropoliski and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393. doi: 10.1007/BF02096763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393. doi: 10.1007/BF02096763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[2]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[3]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[4]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[5]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[6]

Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004

[7]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[8]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[9]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[10]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[11]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[12]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[13]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020449

[14]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[15]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[16]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[17]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[18]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[19]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[20]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (21)
  • HTML views (106)
  • Cited by (0)

Other articles
by authors

[Back to Top]