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On the Reducibility of a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter Near the Equilibrium

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Abstract

In this paper, we consider the following nonlinear analytic quasi-periodic Hamiltonian system

$$\begin{aligned} \dot{x}=(A+\varepsilon Q(t))x+\varepsilon g(t)+h(x,t),~x\in R^{2n}, \end{aligned}$$

where A is a constant matrix with multiple eigenvalues, \(h=O(x^2)(x\rightarrow 0)\), and h(x, t), Q(t) and g(t) are analytic quasi-periodic on \(D_\rho \) with respect to t. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by a quasi-periodic symplectic transformation, Hamiltonian system can be reducible to a quasi-periodic Hamiltonian system with an equilibrium.

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Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Physics, Xuzhou Institute of Technology, Xuzhou, 221111, People’s Republic of China

    Jia Li, Chunpeng Zhu & Shouting Chen

Authors
  1. Jia Li
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  2. Chunpeng Zhu
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  3. Shouting Chen
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Corresponding author

Correspondence to Chunpeng Zhu.

Additional information

The authors are supported by the Scientic Research Foundation of Xuzhou Institute of Technology grant XKY2012302, the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 13KJD110009 and 14KJB110025, NSFC grant by 11301454, 11271168 and 11501234, the Natural Science Foundations of Jiangsu Province grant BK20151160, and the talented person summit in Jiangsu Province grant 2013JY003.

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Li, J., Zhu, C. & Chen, S. On the Reducibility of a Class of Quasi-Periodic Hamiltonian Systems with Small Perturbation Parameter Near the Equilibrium. Qual. Theory Dyn. Syst. 16, 127–147 (2017). https://doi.org/10.1007/s12346-015-0164-x

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