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Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

2004; Elsevier BV; Volume: 202; Issue: 2 Linguagem: Inglês

10.1016/j.jde.2004.03.037

1090-2732

Wiktor Radzki, Sławomir Rybicki,

Nonlinear Differential Equations Analysis

We study connected branches of nonconstant 2 π -periodic solutions of the Hamilton equation x ̇ (t)=λJ ∇ H(x(t)), where λ ∈(0,+∞), H∈C 2 ( R n × R n , R ) and ∇ 2 H(x 0 )= A 0 0 B for x 0 ∈∇ H −1 (0). The Hessian ∇ 2 H ( 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 x ̇ (t)=J ∇ H(x(t)) emanating from x 0 . We describe also minimal periods of trajectories near x 0 .