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Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

W. Radzki·S. Rybicki
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Abstract

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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W. Radzki, S. Rybicki. 2006-04-12. Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems. https://arxiv.org/abs/math/0604288

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