Monotonicity and critical points of the period function for potential system

Abstract: 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.