Monotonicity and critical points of the period function for potential system

This paper is concerned with the analytic behaviors (monotonicity, isochronicity and the number of critical points) of period function for potential system $\ddot{x}+g(x)=0$.We give some sufficient criteria to determine the monotonicity and upper bound to the number of critical periods. The conclusion is based on the semi-group properties of (Riemann-Liouville) fractional integral operator of order $\frac{1}{2}$ and Rolle's Theorem. In polynomial potential settings, bounding the the number of critical periods of potential center can be reduced to counting the real zeros of a semi-algebraic system. From which we prove that if nonlinear potential $g$ is odd, the potential center has at most $\frac{deg(g)-3}{2}$ critical periods. To illustrate its applicability some known results are proved in more efficient way, and the critical periods of some hyper-elliptic Hamiltonian systems of degree five with complex critical points are discussed, it is proved the system can have exactly two critical periods.