The period function for second-order quadratic ODEs is monotone (Q1773862)

Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, C. Chicone conjectured that all centers encountered in the family of second-order quadratic differential equations \(\ddot{x}=V(x,\dot{x}),\) being \(V\) a quadratic polynomial, should have a monotone period function. C. Chicone solved some of the cases but some others remained unsolved. In this paper, the authors fill up these gaps by using a new technique based on the existence of Lie symmetries. This technique can be used to reprove all the cases that have been already solved, providing in this way a compact proof for all second-order quadratic differential equations. They also prove that the property on the period function is no longer true when \(V\) is a polynomial of which a nonlinear part is homogeneous of degree \(n>2.\)