This paper improved the traditional IS-LM macro-economic model.

This paper focused on the Hopf bifurcation, studied the parameter conditions and complex dynamic characteristics.

Introduced the first Lyapunov coefficient to study the stability of the limit cycle, i.e. the stability of business cycle.

The limit cycle remain stable state in most range of parameters; and gradually unstable with the increase of parameters, i.e. the IS-LM system gradually no longer convergence in business cycle.