The period function for second-order quadratic odes is monotone

Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [3] conjectured that all the centers encountered in the family of second-order differential equations ẍ = V (x, x ̇), being V a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [8]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true when V is a polynomial which nonlinear part is homogeneous of degree n > 2.