Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

AbstractWe study connected branches of nonconstant 2π-periodic solutions of the Hamilton equationẋ(t)=λJ∇H(x(t)),where λ∈(0,+∞), H∈C2(Rn×Rn,R) and ∇2H(x0)=A00B for x0∈∇H−1(0). The Hessian ∇2H(x0) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given (x0,λ0). As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system ẋ(t)=J∇H(x(t)) emanating from x0. We describe also minimal periods of trajectories near x0