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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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