New lower bound for the number of critical periods for planar polynomial systems

TL;DR

This paper establishes new lower bounds on the number of critical periods in planar polynomial Hamiltonian systems, improving previous results and providing specific bounds based on the degree of the system.

Contribution

The paper introduces new lower bounds for the number of critical periods in polynomial Hamiltonian systems, doubling the dominant term of previous bounds.

Findings

Lower bound of n-2 for degree n polynomial potential systems.

Lower bounds of n^2/2 + n - 5/2 (odd n) and n^2/2 - 2 (even n) for critical periods.

These bounds are the first of their kind and significantly improve existing results.

Abstract

In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain the lower bounds for the number of critical periods for these systems. For polynomial potential systems of degree $n$, we provide a lower bound of $n-2$ for the number of critical periods, and for polynomial systems of degree $n$, we acquire a lower bound of $n^2/2+n-5/2$ when $n$ is odd and $n^2/2-2$ when $n$ is even for the number of critical periods. To the best of our knowledge, these lower bounds are new, moreover the latter one is twice the existing results up to the dominant term.