The center conditions and local bifurcation of critical periods for a Liénard system (Q632912)

The authors study the two dimensional system \[ \dot x = y,\quad \dot y = -g(x) - f(x) y \] of differential equations that is equivalent to the Liénard system \(\ddot x + f(x) \dot x + g(x) = 0\) in the special case that \(f(x) = a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4\) and \(g(x) = x + b_2 x^2 + b_3 x^3 + b_4 x^4\). They characterize the existence of a center at the origin, its degeneracy, and isochronicity, and count the number of critical periods that can bifurcate from the origin.