Equivalent synchronization of chaos in driven and in autonomous systems.

The synchronization behavior of a network of chaotic elements subject to either an external forcing or a coupling function of their internal variables can be inferred from the behavior of a individual element in the system, which can be seen as a single drive-response system. From the conditions for stable synchronization in this simple drive-response model with minimal ingredients, we find minimal, equivalent conditions for the emergence of complete and generalized chaos synchronization in either driven and autonomous associated systems. We show that the presence of a common drive or a coupling function for all times is not indispensable for reaching synchronization in a system of chaotic oscillators, nor is the simultaneous sharing of a field, either external or internal, by all the elements. In the case of an autonomous system, the coupling function does not need to depend on all the internal variables for achieving synchronization, and its functional form is not crucial for generalized synchronization. What becomes essential for reaching synchronization in a dynamical network is the sharing of some minimal information by its elements, on the average, over long times, independently of the nature (external or internal) of its source. The extension of this autonomous-driven system analogy to other forms of collective behaviors is discussed.