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Separatrix splitting at a Hamiltonian $$0^2i\omega$$ bifurcation. (English) Zbl 1347.37102
The authors consider in a neighborhood of the origin in $$\mathbb{R}^4$$ the generic bifurcation of the equilibrium in one-parameter Hamiltonian systems with two degrees of freedom and relevant real-analytic families of Hamiltonians $$F_{\mu}=H_{\mu}(x_1,x_2,y_1,y_2)$$ under the assumption that the unperturbed fixed point zero has two pure imaginary eigenvalues $$\pm i \omega_0$$ and the non-semisimple double zero eigenvalue. For this example it is studied the splitting of a separatrix in a generic unfolding of a degenerate equilibrium. The corresponding integrable normal forms has a normally elliptic invariant manifold of dimension two on which the truncated normal form has a separatrix loop. The splitting of this loop is exponentially small compared to the small parameter.
The basic result of this article consists in the derivation of the asymptotic expression for the separatrix splitting.

##### MSC:
 37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 70K50 Bifurcations and instability for nonlinear problems in mechanics 70K70 Systems with slow and fast motions for nonlinear problems in mechanics
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