Volume-preserving and volume-expanding synchronized chaotic systems

Many schemes for synchronizing chaotic systems have emerged in recent years. Several of them have been suggested for use in communications. In this paper we pay special attention to communications issues. We show that one can use dynamical systems techniques to produce synchronized, chaotic systems that have features that are desirable in private and secure communications systems. These features are (i) less structure or pattern to the attractor, (ii) very low time correlations, (iii) more than one positive Lyapunov exponent (hyperchaotic), (iv) synchronizable with one transmitted (scalar) signal, (v) rapid synchronization in few time steps, (vi) the ability to add information to the chaotic carrier in a nonlinear fashion, and (vii) easy to make and analyze. We accomplish this by introducing a simple expanding-folding view of $n$-dimensional maps which leads to volume-preserving and volume-expanding maps in many cases whose trajectories cover a large nonzero volume of phase space and which have the desired properties.