Codimension-2 bifurcations of a generalized three-dimensional cubic jerk system

This paper is devoted to analyzing the codimension-2 bifurcations of a generalized three-dimensional (3D) jerk system which is jerk function with all quadratic and cubic terms. First, the locally stability of Bagdanov–Takens (double-zero) equilibrium is systematically investigated using the center manifold and normal form theory combined with some bifurcation theories, and under some certain parameter conditions, the Bagdanov–Takens equilibrium of this jerk system could be a saddle, a focus, a degenerate equilibrium or an equilibrium which neighborhood is formed from an elliptic sector and a hyperbolic sector on its two-dimensional center manifold. Second, some adequate parameter conditions for guaranteeing the occurrence of Hopf bifurcation, Bautin (degenerate Hopf) bifurcation and the stability of the Hopf equilibrium of this jerk system are given. Finally, numerical simulation is used to verify the above theoretical analyses.