Bautin bifurcation for the Lengyel–Epstein system

Abstract

In this paper, complete analysis is presented to study Bautin bifurcation for the Lengyel–Epstein System $$\begin{aligned} \left\{ \begin{array}{l} \frac{du}{dt}=a-u-\frac{4uv}{1+u^2},\\ \frac{dv}{dt}=\sigma b\left( u-\frac{uv}{1+u^2}\right) . \end{array}\right. \end{aligned}$$ Sufficient conditions for $$a$$ and $$b$$ are given for the system to demonstrate Bautin bifurcation. By using $$b$$ and $$\alpha =a/5$$ as bifurcation parameters and computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal form with unfolding parameters is derived to obtain the bifurcation diagrams such as Hopf and double limit cycle bifurcations. An example is given to confirm that the system has two limit cycles.