Synchronization and
control in small assemblies of chaotic systems
A-Iglesias 
, J.M.-Gutiérrez , J.-Güémez 
and M.A.-Matías 
Dpt. of Applied Mathematics and Comp. Sciences, University of
Cantabria, Spain
Dpt. of Applied Physics, University of Cantabria, Spain
Dpt. of Chemical-Physics, University of Salamanca, Spain
Recently, a new control chaos method able to stabilize chaotic systems by introducing small perturbations in the system variables has been suggested [1] and several features of its applications to discrete [2] and continuous [3-4] dynamical systems has been shown.
On the other hand, several recent studies have shown the possibility of synchronizing chaotic systems. In particular, Pecora and Carroll [5] consider the situation of undirectional coupling, in which one has a drive-response couple, although other extensions of this method have already described in the literature, as in [6], where a modification of the previous method allowsus to reproduce the driving signal with single connection, increasing, thus, the number of potential connections of a given system.
In this work, we analyze the interesting effects obtained when both methods are simultaneously applied to small networks of Chua's circuits [7]. We stabilize a Chua's circuit (which acts as drive of the cascade) by applying the above introduced control chaos method, and its regular behavior induces, by synchronizing with other circuits of the cascade, a regular behavior in the network. This synchronization has been obtained by introducing the driving signal at a given place of the evolution equations of the response (see [6]). As consequence, it is possible to regenerate the input signal within a single connection. It implies the possibility of using twice the same connection without alternating with other one and, therefore, a number of different networks with different connectivities can be set up. In particular, a few different connections have been considered for the case offour oscillators in a linear geometry by using a single connection. Another connection can be carried out through a closed loop, either with alternating stable connections or being one of them unstable (from the view point of synchronization), where the stability of the synchronized state can be written on terms of the corresponding transverse Lyapunov exponents.
By conclusion, the possible uses of this method include controlling chaos in spatiotemporal systems involving phenomena whose description cannot be captured by a low-dimensional dynamical model, through extended arrays of coupled chaotic elements (for example, Chua's circuits) in which several connections among the different units coexist or the possibility of cascading a large number of low-dimensional systems with different connections without reducing the dimensionality of the response systems. This kind of connection can be applied to the case where the units represent model neurons. Then, arrays of circuits might be useful as information proccesing systems by sychronizing one to each other, mimicking the behavior observed in physiological studies.