an:01038819
Zbl 0874.34044
Iooss, G??rard
Existence of orbits homoclinic to an elliptic equilibrium, for a reversible system
FR
C. R. Acad. Sci., Paris, S??r. I 324, No. 9, 993-997 (1997).
00041254
1997
j
34C37 37C80 37G05
reversible vector field in \(\mathbb{R}^ 4\); homoclinic; pair of double non semisimple pure imaginary eigenvalues; two reversible orbits homoclinic; fixed point argument
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.