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