The period functionsʼ higher order derivatives

We prove a formula for the n-th derivative of the period function T in a period annulus of a planar differential system. For n=1, we obtain Freire, Gasull and Guillamon formula for the periodʼs first derivative (Chicone and Dumortier, 1993) [17]. We apply such a result to Hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order ODEʼs to be convex.