In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields that there exist at most seven small amplitude limit cycles restricted to one center manifold and Hopf cyclicity 8 at the positive equilibrium. Secondly, by using the normal form algorithm, we discuss the existence of stable periodic solution via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which is rarely considered. Finally, some related illustrations are given by means of numerical simulation.

In this paper, the multiple bifurcation of limit cycles for a segmented disc dynamo system is studied. The formal series method for calculating the singular point quantities is applied to determine the highest order focus value at Hopf bifurcation point. For two cases of the segmented disc dynamo system, namely the system with or without friction coefficient (abbr. SDDF- or SDD-model), the maximum number of limit cycles is obtained at the symmetrical equilibrium points under the condition of synchronous perturbation respectively. At the same time, the parameters condition is classified for exact number of limit cycles near each weak focus. Finally, we find that all equilibrium points of the heart model are Jacobi unstable under certain parameter values.