Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

Details Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

2006-04-12

arxiv.org/abs/math/0604288v1

Journal of Differential Equations 202(2) (2004), 284-305

Mathematics

Classical Analysis and ODEs

18 pages

Scientific paper

We study connected branches of non-constant {$2\pi$-pe}riodic solutions of the Hamilton equation \begin{displaymath} \dot{x}(t)=\lambda J\nabla H(x(t)), \end{displaymath} where $\lambda\in\halfline,$ $H\in C^2(\R^n\times\R^n,\R)$ 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 $(x_0,\lambda_0).$ 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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