Monotonicity and critical points of the period function for potential system

TL;DR

This paper analyzes the behavior of the period function in potential systems, providing criteria for monotonicity and bounds on critical points using fractional calculus and algebraic methods, with applications to polynomial and hyper-elliptic systems.

Contribution

It introduces new criteria for monotonicity and bounds on critical points of the period function, utilizing fractional integrals and algebraic techniques, and applies these to polynomial and hyper-elliptic systems.

Findings

Maximum of (deg(g)-3)/2 critical periods for odd polynomial potentials

Simplified proofs of known results on critical periods

Exact two critical periods in certain hyper-elliptic Hamiltonian systems

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.