We discuss three examples of bimolecular mass-action systems with species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants then exhibits stable limit cycles, but no chaotic behaviour. For some in the Feinberg--Berner system, equilibrium, an unstabe cycle, cycle coexist. All networks are minimal sense. By ...

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 smal...

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the norm...

A part of the well-known Hilbert’s 16th problem is to consider the existence of maximal number of limit cycles for a general planar polynomial system. In general, this is a very difficult question and it has been studied by many mathematicians (see e.g. [Bautin, 1952; Zhang et al., 2004]). By [Ye, 1986] we know that there exists a quadratic system having four limit cycles. [Bautin, 1952] proved...

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional systemwhen the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurca...