Converting a General 2D Quadratic Autonomous System to a 2D Lorenz-Type System

Under three necessary conditions for preserving the essential qualitative properties of the 3D Lorenz system, a general 2D quadratic autonomous system is converted to a 2D Lorenz-type system (2DLTS). A canonical form of the 2DLTS is derived with aid of a normalization technique. It is found that the 2DLTS can be converted to the 2D Duffing oscillator model under certain conditions. Furthermore, it is shown that the 2DLTS undergoes pitchfork bifurcation and Hopf bifurcation. Finally, approximate periodic solutions of both the 2DLTS near the Hopf bifurcation point and a time-periodically forced system are obtained.