Quadratic Systems with Center and Their Perturbations

We study the quadratic systems on a plane = (i + λ)z + Az2 + B z2 + C 2, z = x + iy. We give a simple algebraic proof of the center conditions, λ = 0 and B = 0 or A = − , B = 1 or A = , B = 1, C = or A = 2, B = 1, C = 1, and of the theorem of Bautin that the number of small-amplitude limit cycles bifurcating from a center or a focus is not greater than 3. We present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case. Here one mistake in Bautin′s work is revealed and some questions concerning focus numbers are answered. We study also global (not small) limit cycles for a small perturbation of the center case with two invariant lines (the Lotka-Volterra system) using Abelian integrals. We show that the number of zeroes of the corresponding Abelian integral is 0, 1, or 2. Some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two-fold center (when two center conditions hold). The problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients.