Local Bifurcations and a Survey of Bounded Quadratic Systems

This paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems. It is shown that the only finite-codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle-node and the Hopf–Takens bifurcations of codimensions 1 and 2 and the Bogdanov–Takens bifurcations of codimensions 2 and 3; furthermore, it is shown that whenever a bounded quadratic system has one of these critical points, then a full generic unfolding of the critical point exists in the class of bounded quadratic systems. Finally, we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system.