Coupled dynamical systems are constructed from simple, low-dimensional dynamical systems and form new and more complex organizations. The possibility of chaotic systems oscillate in a coherent and synchronized way is not an obvious phenomenon, since it is not possible to reproduce exactly the initial conditions and infinitesimal perturbations of the initial conditions lead to the divergence of nearby starting orbits. However, the chaotic dynamics introduces new degrees of freedom in ensembles of coupled systems: when chaotic oscillators are coupled and synchronization is achieved, in general the number of dynamic degrees of freedom for the coupled system effectively decreases.
In order to obtain identical and generalized synchronization, we apply various unidirectional and bidirectional coupling schemes between Lorenz or Rössler systems, with control parameters which lead to chaotic behavior, including the hyperchaotic Rössler system. We combine some of these with total or partial substitution on the linear terms of the second system. In some cases we only conclude about local stability of the synchronous state, but we present coupling schemes where the global stability is guaranteed. The conditions of global stability are obtained from a different approach of the Lyapunov direct method for the transversal system.
Our motivation for researching chaos synchronization methods is to explore their practical application in various scientific areas, such as physics, biology or economics. The ability of nonlinear oscillators to synchronize with each other is a basis for the explanation of many real processes and synchronization of chaos is thus a robust property expected to hold in mademan devices and plays a significant role in science.