On a structural property for a class of chaotic systems

We consider a simple class of autonomous nonlinear systems, which includes several known chaotic and hyperchaotic dynamics. We first note that a system belonging to this class can be obtained from a linear system by means of a nonlinear state feedback. Then we show that a necessary condition for the existence of a nonlinear feedback generating chaotic dynamics is that the uncontrollable eigenvalues of the linear system, if any, must be stable. This result makes a contribution to the emerging issue of designing new chaotic systems. Finally, we give a unified framework to synchronize a class of autonomous as well as a class of nonautonomous chaotic or hyperchaotic systems via a scalar signal.