Normal forms with exponentially small remainder: application to homoclinic connections for the reversible \(0^{2+}i\omega\) resonance
type: Item
instance of: scholarly article
title: Normal forms with exponentially small remainder: application to homoclinic connections for the reversible \(0^{2+}i\omega\) resonance
zbMATH Open document ID: 1066.34041
author: Gérard Iooss
author: Eric Lombardi
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-23T00:00:00Z
review text: 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.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 37C29
zbMATH DE Number: 2137981
zbMATH Keywords: reversible analytic vector fields
zbMATH Keywords: resonant fixed point
zbMATH Keywords: homoclinic connection
zbMATH Keywords: normal form
zbMATH Keywords: exponentially small periodic orbits
MaRDI profile type: Publication
full work available at URL: j.crma.2004.10.002
OpenAlex ID: W1982437849
cites work: Q4266071
cites work: Polynomial normal forms with exponentially small remainder for analytic vector fields
cites work: Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems
DOI: 10.1016/J.CRMA.2004.10.002
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description: scientific article; zbMATH DE number 2137981