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

TL;DR

This paper extends Chicone's criterion to determine when the period function of planar Hamiltonian systems is monotonic, providing explicit conditions and analyzing polynomial systems with parameters.

Contribution

It generalizes Chicone's classical criterion to a broader class of Hamiltonian systems, offering explicit conditions for period function monotonicity.

Findings

Established sufficient conditions for monotonicity of the period function.

Extended Chicone's criterion to a wider Hamiltonian framework.

Analyzed polynomial Hamiltonian systems with parameter-dependent period behavior.

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 $x' = y$, $y' = -F'(x)$, 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 $a, b, c \in \mathbb{R}$.