Quasi-conservative Maps And Normal Forms For Unstable Fixed Points

The real eigenvalues λ1 and λ2 of an unstable fixed point of a plane diffeomorphism are resonant when λj = λn 1λm 2. To avoid the presence of dense resonances in a one-parameter family of maps we propose a generalisation of the Birkhoff normal form for quasi-conservative maps. This generalisation does not converge on the resonances but even there it can be taken as an excellent approximation. We use it to calculate homoclinic points with great precision. © 1994.18513845Gustavson, (1966) The Astronomical Journal, 21, p. 670Moser, (1956) Communications on Pure and Applied Mathematics, 9, p. 673da Silva Ritter, de Almeida, Douady, (1987) Physica D, 29, p. 181Arnold, (1980) Chapitres supplémentaires de la théorie des équations différentielles ordinaires, , Mir, MoscowGuckenheimer, Holmes, (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, , Springer, BerlinBirkhoff, (1927) Acta Math., 43, p.