自治哈密顿系统周期解的退化歧点
Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems
摘要：
研究了Hamilton方程\begin{displaymath} \dot{x}(t)=λJ\nabla H(x(t)), \end{displaymath}的非常数{ $2π$ -周期}周期解的连通分支，其中$λ\in\halfline,$ $H\in C^2(\R^n\times\R^n,\R)$和$ \displaystyle \nabla^2H(x_0)= [ \begin{array}{cc} A&0 0&B \end{array} ] $对于$x_0\in\nabla H^{-1}(0).$，Hessian $\nabla^2H(x_0)$可以是奇异的.我们给出了从给定的$(x_0,λ_0).$分支存在的充分条件，从而证明了关于从$x_0.$分支出的Hamilton系统$\dot{x}(t)=J\nabla H(x(t))$的任意周期非平稳轨线的连通分支的存在性定理，并描述了在$x_0.$附近轨线的极小周期
英文摘要：
We study connected branches of non-constant {$2π$-pe}riodic solutions of the Hamilton equation \begin{displaymath} \dot{x}(t)=λJ\nabla H(x(t)), \end{displaymath} where $λ\in\halfline,$ $H\in C^2(\R^n\times\R^n,\R)$ and $ \displaystyle \nabla^2H(x_0)= [ \begin{array}{cc} A&0 0&B \end{array} ] $ for $x_0\in\nabla H^{-1}(0).$ The Hessian $\nabla^2H(x_0)$ can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given $(x_0,λ_0).$ As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system $\dot{x}(t)=J\nabla H(x(t))$ emanating from $x_0.$ We describe also minimal periods of trajectories near $x_0.$
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