Planar S-systems: Global stability and the center problem

TL;DR

This paper investigates the global stability of positive equilibria and the center problem in planar S-systems, a class of power-law dynamical systems, and constructs an example with two limit cycles.

Contribution

It provides new results on stability and the center problem for planar S-systems and constructs a specific system exhibiting two limit cycles.

Findings

Proved global stability conditions for positive equilibria.

Solved the center problem for planar S-systems.

Constructed an example with two limit cycles.

Abstract

S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a planar S-system with two limit cycles.