New insights to the Hide-Skeldon-Acheson dynamo (Q2081923)

The author considers the Hide-Skeldon-Acheson dynamo model \[ \begin{aligned} & \dot{x}=x(y-1)-\beta z, \\ & \dot{y}=\alpha (1-{{x}^{2}})-ky, \\ & \dot{z}=x-bz;\quad (x,y,z)\in {{\mathbb{R}}^{3}}, \end{aligned} \] where \(a,\beta ,b,k\in \mathbb{R}\) are system parameters. For positive values of the parameters, conditions for locally asymptotic, as well as globally uniform and asymptotic stability of equilibrium are found. The boundedness of all orbits of the system as well by using the generalized Melnikov method, the existence of periodic orbits is proved. The author analyze the occurrence of the Hopf bifurcation and gives formulas for determining the direction, stability and period of bifurcating periodic solutions. Further, the author considers the entire range of possible values of the parameters and gives numerical examples of hidden periodic and hidden chaotic attractors.