Quadratic double centers and their perturbations

This article begins with a full description of the quadratic planar vector fields which display two centers. We follow the method proposed by Chengzhi Li and provide more detailed analysis of the different types of double centers using the classification: Hamiltonian, reversible, Lotka-Volterra, Q4, currently used for centers of quadratic planar vector fields. We also describe completely the different possible phase portraits and their Poincaré compactification. We show that the double center set is a semi-algebraic set for which we give an explicit stratification (see figure 2). Then we initiate a study of the perturbations within quadratic planar vector fields of the most degenerated case which is the double Lotka-Volterra case. The perturbative analysis is made with the method of successive derivatives of return mappings. As usual, this involves relative cohomology of the first integral which is in that case a rational function. In this case, we have to deal with a kind of “relative logarithmic cohomology” already known in singularity theory. We succeed to compute the first bifurcation function by residue techniques around each centers and they differ from one center to the other.