\(3\)-dimensional Hopf bifurcation via averaging theory of second order
The authors consider the system \[ \begin{aligned}\dot{x}&=(a_{1} \varepsilon + a_{2} \varepsilon^{2})x - by +f_{1}(x,y,z) + \varepsilon g_{1}(x,y,z) \\ \dot{y} &= bx + (a_{1} \varepsilon + a_{2} \varepsilon^{2} )y + f_{2}(x,y,z) + \varepsilon g_{2}(x,y,z) \\ \dot{z} &= (c_{1} \varepsilon + c_{2} \varepsilon^{2}) z + f_{3}(x,y,x)+ \varepsilon g_{3}(x,y,z) ,\end{aligned} \] where \(f_{i}\) and \(g_{i}\) are quadratic polynomial functions in \(x, y, z\). The method of second order averaging is used to show that at most three limit cycles can bifurcate from the origin when \(\varepsilon = 0\). An example of a system with three limit cycles bifurcated from the origin is also given.
scientific article; zbMATH DE number 5648560