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This paper is concerned with the analytic behaviors (monotonicity,isochronicity and the number of critical points) of period function forpotential system $\ddot{x}+g(x)=0$.We give some sufficient criteria todetermine 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. Inpolynomial potential settings, bounding the the number of critical periods ofpotential center can be reduced to counting the real zeros of a semi-algebraicsystem. From which we prove that if nonlinear potential $g$ is odd, thepotential center has at most $\frac{deg(g)-3}{2}$ critical periods. Toillustrate its applicability some known results are proved in more efficientway, and the critical periods of some hyper-elliptic Hamiltonian systems ofdegree five with complex critical points are discussed, it is proved the systemcan have exactly two critical periods.

Monotonicity and critical points of the period function for potential system