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

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 $x^{\mathrm{\prime }}=y$, $y^{\mathrm{\prime }}=-F^{\mathrm{\prime }}(x)$, 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 $$H(x,y)=(1\mathrm{/}2)x^2+(a\mathrm{/}3)x^3+(b\mathrm{/}4)x^4+(1\mathrm{/}2)y^2+(c\mathrm{/}4)y^4,$$ as a function of the parameters $a,b,c\in \mathbb{R}$.