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On stability at the Hamiltonian Hopf bifurcation. (English) Zbl 1229.37056

Summary: For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.

MSC:

37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
70H08 Nearly integrable Hamiltonian systems, KAM theory
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