Local bifurcations and a survey of bounded quadratic systems

This paper presents a survey of the known results for bounded quadratic sys tems as well as a study of the local bifurcations that occur at critical po ints of such systems. It is shown that the only finite-codimension bifurcat ions that occur at a critical point of a bounded quadratic system are the s addle-node and the Hopf-Takens bifurcations of codimensions 1 and 2 and the Bogdanov-Takens bifurcations of codimensions 2 and 3; furthermore, it is s hown that whenever a bounded quadratic system has one of these critical poi nts, then a full generic unfolding of the critical point exists in the clas s of bounded quadratic systems. Finally, we give a complete list of those l imit 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 lim it cycles that can occur in a hounded quadratic system. (C) 2000 Academic P ress.