A new Chebyshev criterion and its application to planar differential systems

TL;DR

This paper introduces a new Chebyshev criterion for integrals, enabling the discovery of new Chebyshev families and providing a simpler, unified approach to analyze limit cycles in planar differential systems.

Contribution

It establishes a novel Chebyshev criterion, constructs new Chebyshev families, and applies these to resolve a conjecture and analyze planar differential systems more effectively.

Findings

Resolved the conjecture posed by Gasull et al. in 2015.

Developed a simpler, unified method for estimating limit cycles.

Constructed several new Chebyshev families.

Abstract

This paper establishes a new Chebyshev criterion for some family of integrals. By virtue of this criterion we obtain several new Chebyshev families. With the help of these new families we can answer the conjecture posed by Gasull et al in 2015. %[J. Differential Equations, 258 (2015), 3286--3303]. Their applications to other two planar differential systems also show that our approach is simpler and in a unified way to handle many kinds of planar differential systems for estimating the number of limit cycles bifurcating from period annulus.