Normal forms with exponentially small remainder: application to homoclinic connections for the reversible \(0^{2+}i\omega\) resonance
This note is devoted to the study of \(S\)-reversible, analytic one parameter families of vector fields in \(\mathbb R^4\): \[ \frac{du}{dx}=\mathcal V (u,\mu ),\quad u\in\mathbb R^4, \mu\text{ is a small parameter},\text{ and } \mathcal V (Su,\mu )= -S\mathcal V (u,\mu ),tag1 \] where \(S\in GL_4(\mathbb R)\) is some reflection. It is assumed that the origin is a \(0^2i\omega\) resonant fixed-point of the family, i.e., \(\mathcal V (0,\mu )=0\) and that the spectrum of the differential \(D_u\mathcal V (0,0)\) is \(\{\pm i\omega ,0\}\) with \(\omega >0\) and \(0\) is a double nonsemi-simple eigenvalue. System (1) is supposed to admit a nondegenerate \(0^{2+}i\omega\) resonance. It is shown how the theorem on normal form with exponentially small remainder recently proved by G. Ioos and E. Lombardi, enables one to prove the existence of homoclinic connections to exponentially small periodic orbits of system (1). Although the existence of homoclinic connections can be obtained by complexification of time, i.e., on the study of the singularities of the solutions of (1) in the complex field [\textit{E.~Lombardi}, Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems, Lecture Notes in Mathematics. Berlin: Springer 1741 (2000; Zbl 0959.34002)], the proof proposed in this note is more simple and appears to work for higher resonances when one studies the existence of homoclinic connections to tori.
scientific article; zbMATH DE number 2137981