On the Reducibility of a Class of Nonlinear Periodic Hamiltonian Systems with Degenerate Equilibrium

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.

Appendix

Lemma 6.1

Consider the following equation of the matrix

$$\begin{aligned} \dot{P}=AP-PA+R(t), \end{aligned}$$

(6.1)

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

$$\begin{aligned} \dot{P}_J=A_J JP_J-P_JJA_J+R_J. \end{aligned}$$

(6.2)

The Eq. (6.2) is changed into

$$\begin{aligned} (\dot{P}_J)^T=A_J J(P_J)^T-(P_J)^T JA_J+R_J. \end{aligned}$$

Since the solution of (6.2) is unique, we have that $(P_J)=(P_J)^T.$$\square$