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Existence of orbits homoclinic to an elliptic equilibrium, for a reversible system. (Existence d’orbites homoclines à un équilibre elliptique, pour un système réversible.) (French) Zbl 0874.34044
Summary: We consider a reversible vector field in $$\mathbb{R}^4$$, where the origin is a critical point, and where the differential at the origin has a pair of double non semisimple pure imaginary eigenvalues $$\pm i\omega$$. We assume that the coefficient $$\varepsilon$$ of a cubic term of the normal form is positive and close to 0, and that a certain coefficient of order 5 is negative. Then we show that there exist two reversible orbits homoclinic to the origin, of size $$\sqrt\varepsilon$$ and such that they oscillate with a damping in $$1/t$$ when $$t$$ tends towards $$\pm\infty$$. For obtaining such a result, we give explicitly the inverse of the linearized operator around the reversible homoclinics of the normal form, and solve the problem by a fixed point argument.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37C80 Symmetries, equivariant dynamical systems (MSC2010) 37G05 Normal forms for dynamical systems
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