Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

摘要

In this paper, we consider the number of zeros of Abelian integral for the system (x) over dot = -y(ax(2) + bx + 1) + epsilon P-n(x, y), (y) over dot = x(ax(2) +bx +1) + epsilon Q(n)(x, y), where a not equal 0, b(2) - 4a 0, and P-n, Q(n) are arbitrary polynomials of degree n. We obtain that H(n) = 2[n+1/2] - 1 if b = 0 and 2[n+1/2] + [n/2] = H(n) = 3[n+1/2] if b not equal 0, where H(n) is the maximuin number of limit cycles bifurcating from the period annulus up to the first order in epsilon. So, the bounds for b = 0 or b not equal 0, n = 2k, k is an element of N are exact.