Monotonicity of the period function for planar Hamiltonian vector fields: A Generalization of Chicone's Criterion

Abstract

This paper investigates the monotonicity of the period function associated with planar Hamiltonian systems of the form $H(x,y)=F(x)+G(y)$. We establish sufficient conditions ensuring the monotonicity of the period function corresponding to a nondegenerate center, expressed explicitly in terms of the functions F and G. Our approach extends Chicone's classical criterion, originally formulated for systems of the type $x^{\prime }=y$, $y^{\prime }=-F^{\prime }(x)$, to a broader Hamiltonian framework. As a main application, we analyze the monotonicity of the period function associated with the center at (0,0) of the polynomial Hamiltonian system $$H(x,y)=(1/2)x^2+(a/3)x^3+(b/4)x^4+(1/2)y^2+(c/4)y^4,$$ as a function of the parameters $a,b,c\in \mathbb{R}$.