Lower bounds for the cyclicity of centers of quadratic three-dimensional systems

TL;DR

This paper investigates the cyclicity of centers in quadratic three-dimensional systems with a Hopf point, establishing lower bounds for bifurcations of limit cycles, including a new bound of 12 for a jerk system.

Contribution

It provides new lower bounds for the cyclicity of centers in quadratic 3D systems and introduces an example with a record bifurcation of 12 limit cycles.

Findings

Lower bounds for cyclicity in known systems like Rossler, Lorenz, Moon-Rand.

A new lower bound of 12 limit cycles bifurcated from a center in a jerk system.

Charting of cyclicity for quadratic three-dimensional systems.

Abstract

We consider quadratic three-dimensional differential systems having a Hopf singular point. We study the cyclicity when the singular point is a center on the center manifold using higher order developments of the Lyapunov constants. As a result, we make a chart of the cyclicity by establishing the lower bounds for several known systems in the literature, among them the Rossler, Lorenz and Moon-Rand systems. Moreover, we obtain an example of a jerk system for which is possible to bifurcate 12 limit-cycles from the center, which is a new lower bound for three-dimensional quadratic systems.