Recent work using chaotic signals to drive nonlinear systems shows that chaotic dynamics is rich in new application possibilities. Among these are stable system design and synchronization.
New Driving Signals
Driven systems are easily visualized as dynamical systems which have as one of their input parameters a dynamical variable from another, often autonomous, dynamical system. We often refer to the source of the driving signal as the drive system and to the driven system as the response system. This can be viewed as the drive sending a signal to the response which then alters its behavior according to the signal. Typically, when driven systems are studied or engineered the driving signals come from constant forces or sine wave forcing. The use of signals from a chaotic system to drive a nonlinear system offers a new type of driving signal.
In our approach [1,2,3,4] two major themes stand out. One is the idea of stability as generalized to chaotic systems. Another is the use of a constructive approach to building useful, chaotically driven systems. We cut apart, duplicate, and paste together nonlinear dynamical systems. Many things can be done with some guidance from what is now known in nonlinear dynamics.
We first examine stability.
Stability of Chaotically Driven Systems
Consider a general «-dimensional, nonlinear response system, w = h(w,v), where the w=dw/dt, the driving signal v is supplied by a chaotic system and w and h are «-dimensional vector functions. The question of stability arises when we ask: given a trajectory w(t) generated by this system for a particular drive v, when is w(t) immune to small differences in initial conditions, i.e. when is the final trajectory unique, in some sense? Fig. 1 shows this schematically.