Abstract
In this paper, we prove the existence of periodic solutions of a class of Hamiltonian systems with degenerate equilibriums under small nonlinear periodic perturbations. Actually we prove that the periodic Hamiltonian systems with small perturbations can be reducible to a periodic Hamiltonian system with an equilibrium by a periodic symplectic mapping. This result is a reformulation of the result in Lu and Xu (Nonlinear Differ Equ Appl 21:361–370, 2014) in the case of Hamiltonian systems.
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The author is supported by the Natural Science Foundations for Colleges and Universities in Jiangsu Province Grant 18KJB110029.
Appendix
Appendix
Lemma 6.1
Consider the following equation of the matrix
where A and R(t) are analytic periodic Hamiltonian matrices. If there exists a unique analytic periodic solution P(t), then the solution P(t) is also Hamiltonian.
Proof
Since A and R are Hamiltonian, let \(A=JA_J\) and \(R=JR_J\), where \(A_J\) and \(R_J\) are symmetric. Let \(P_J=J^{-1}P\). If \(P_J\) is symmetric, then P is Hamiltonian. Now we prove that \(P_J\) is symmetric. The Eq. (6.1) becomes
The Eq. (6.2) is changed into
Since the solution of (6.2) is unique, we have that \((P_J)=(P_J)^T.\)\(\square \)
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Li, J. On the Reducibility of a Class of Nonlinear Periodic Hamiltonian Systems with Degenerate Equilibrium. Qual. Theory Dyn. Syst. 19, 63 (2020). https://doi.org/10.1007/s12346-020-00396-z
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DOI: https://doi.org/10.1007/s12346-020-00396-z