Synchronization in the Lorenz system: Stability and robustness

We present necessary and sufficient conditions for the existence of synchronization in a class of continuous-time nonlinear systems: the so-called -affine systems. We apply the results to the Lorenz attractor. The robustness of the synchronization against parameter value variations is discussed using the Lyapunov stability theory for perturbed systems. We obtain sufficient conditions that guarantee a bounded steady-state error. This technique gives conservative results; however, in some systems like that of Lorenz, it provides definitive results about the existence of the synchronization. Furthermore, we give estimates of the maximal error as a function of the difference between the parameter values of the systems to be synchronized.