Oscillations and chaos in simple quadratic systems

This paper is concerned with the study of third order quadratic and autonomous systems and the interest is oriented to the stable periodic oscillations. From the "jerk" equation model, the classes of minimal complexity presenting a Hopf bifurcation are derived and their local characterization is carried out by means of a suitable harmonic balance technique. Other possible system reductions preserving the oscillations are studied and the numerical analysis confirms the obtained results. Some bifurcations and related routes to chaos are also exhibited by these simple systems. A comparison with previous results on the subject is also presented.