On the Convergence and Behavior of Three Dimensional Normal Forms

Low codimensional bifurcations of control systems have been classified using the theory of normal forms. However, the normal form theory was developed as local results. To understand the behavior of control systems outside a local area of a bifurcation point, it is necessary to study the convergence of the normal forms if the vector fields are analytic. The convergence is a long time open problem. In this paper, we address the convergence problem of normal forms for three dimensional control systems. Bounded attractors are found for a family of such normal forms, which include the controlled Lorenz system as a special case.