Realizability of the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation in a vector field

Abstract

In this paper, a new and efficient mechanism to compute the normal forms for 1:1 resonant Hopf bifurcation is developed. For a vector field given by ordinary differential equations, by assuming that eigenvalues at an equilibrium point are purely imaginary, double, and non-semisimple, the mechanism provides a direct method to calculate the coefficients for the normal forms. In particular, we present the following results: (1) a simple direct method to determine a basis of the complementary spaces for the Lie transform; (2) a simple direct method to determine the projection of any vector in H43 to the complementary spaces; and (3) the normal forms for vector fields with double purely imaginary eigenvalues with geometric multiplicity one. In addition, explicit formulas for coefficients with three unfolding parameters are obtained and the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation are obtained.