Saddle quantities and applications

Abstract

In this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation (GHB ∗) and the practical computational aspects. For this purpose we first compare the Dulac normal form with the Joyal normal form. These forms were both used to prove the GHB ∗ theorem. But the second one is far more practical from the algorithmic point of view. We then show that the information carried by these normal forms can be computed in a much simpler way, using what we shall call dual Lyapunov constants. The coefficients of a normal form or the dual Lyapunov quantities are particular cases of what we shall call saddle quantities. We calculate the saddle quantities for quadratic systems, and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation. We also study the homoclinic loop bifurcation of order 5, appearing in a 6-parameter family close to a Hamiltonian system. To our knowledge, this is the first time that one can find a complete description of a GHB ∗ of such high order. Finally we calculate the saddle quantities for a symmetric cubic vector field, and we deduce a bound for the number of limit cycles that appear in a GHB ∗ .