The study of linear systems has contributed to advances in the theory of control and our understanding of a wide variety of physical, chemical, biological, and even social, phenomena. Nevertheless, the great majority of systems are nonlinear and it is necessary to resort to new techniques to extract information from them and gain a better understanding of their behavior and evolution. In some cases this allows us to predict the future of the system, in other it allows for control of it, an yet in others it permits us to adapt their behavior to suit some special purpose.
In this paper we have described with some detail the applications of chaotic dynamical systems in cryptography and estimation. In cryptography, we have seen how it is possible to generate efficient cipherers using, for example, a quadratic map exhibiting fully developed chaos. Variations on this theme are easy to come by with, in fact, instead of resorting to a substitution scheme one might use a chaotic system to generate a permutation matrix and use this to encode a message .
In estimation, the usefulness of the proposed scheme depends on the appropriate selection of the probability table that defines what we called a confidence function and the dynamical systems we choose to conform our iterated function system. By observing the system it is reasonable to gain a better understanding of its operation so that predictions can be made almost up to the limit imposed by the prediction horizon.
Nonlinear analysis techniques allow us, in principle, to distinguish between purely random sequences (noise) and complex sequences generated by a chaotic dynamical system. Chaos, as we have seen, is deterministic and its predictability is limited only by our capacity to make exact measurements and to perform exact arithmetic.
In a way, the fact that a system is nonlinear imposes a limit on how much we can now about the system. This reminds us of Heisenberg's uncertainty principle, in which incompatible operators limit our simultaneous knowledge of the variables involved. But whereas the uncertainty principle is of purely physical origin, our prediction horizon is a physical and computational result.
There are more ways to exploit chaotic system's characteristics. We hope the ones we have proposed might stimulate the imagination and encourage further research in this topic.