Asymptotic behaviour and existence of a limit cycle of cubic autonomous systems

In this paper a 2-dimensional real autonomous system with polynomial right-hand sides of a concrete type is studied. Hopf bifurcation is analysed and existence of a limit cycle is proved. A new formula to determine stability or unstability of this limit cycle is introduced. A positively invariant set, which is globally attractive, is found. Consequently, existence of a stable limit cycle around an unstable critical point is proved and also a sufficient condition for non-existence of a closed trajectory in the phase space is given. Global characteristics of the system are studied. An application in economics to the dynamic version of the neo-keynesian macroeconomic IS-LM model is presented.