In the late 1800's, Henri Poincaré won a prize for his essay on the famous n-body problem. The original problem asked if it was possible to solve explicitly the orbits in space followed by n massive objects, acting on gravitational attraction alone, if their positions and momenta were known at some instant in time. Poincaré , assuming the validity of Newtonian mechanics, could not find an explicit solution (nobody has solved the problem for more than two objects). Yet, he did find a collection of interesting properties of such time-evolving systems. He found libration points, quasi-stable regions, and open (aperiodic) trajectories. In doing so he inaugurated modern research on chaotic systems. Incidentally, NASA makes use of these kind of strange ``orbits`` to send probes to other planets expending minimal energy. So we might as well say that the implications of this area of research are literally ``out of this world``.
Research in chaotic systems aims to understand how a deterministic dynamical system might exhibit chaotic behavior, the kind of systems capable of this behavior, the ways available to control it, and the practical and theoretical implications that follow.
Perhaps the first scientific paper reporting observed chaotic behavior in a relatively simple dynamical system is one published in 1963 due to Edward N. Lorenz [1] . In it, a series of computer simulations of a simplified climatological model would yield very different predictions when the initial conditions were altered, even if it only was by a very small amount. Lorenz argued that uncertainties in the initial conditions and the characteristics of the solutions of this system would invalidate any prediction after a stretch of time. In other words, there was no sense in extrapolating too far into the future because initial errors, no matter how small, would amplify in time up to the point where the uncertainty in the prediction and the prediction itself would be of comparable magnitude.
Therefore, there is a limit to how much we can predict a phenomenon and this depends on how fast do nearby solutions (solutions starting at slightly different initial conditions) diverge. This establishes what is known as the predictability horizon: the time into the future after which any prediction is hopeless. If we reduce the initial uncertainty it is possible to increase the horizon, but unlike Lorenz thought, even under no initial uncertainty the predictability horizon does not approach infinity. The reason stems from the fact that dynamical systems are simulated in a computer using finite precision arithmetic. Every operation introduces a small round-off error and these errors propagate with every calculation. Of course, one might abandon floating point arithmetic and implement mathematical subroutines that keep track of every bit that is being generated. Unfortunately, this cannot be sustained for too long since it imposes an exponential demand of computer memory and computational time, so eventually one must round-off a result and introduce an error [2].
Thus, we must abandon the idea of predicting perfectly the evolution of a chaotic system. However, this does not imply that all prediction efforts are in vane, or that we cannot make valuable observations that will give us insight into the phenomenon at hand. Quite the contrary, even when the predictability horizon imposes a limit to our prediction hopes, this limit is often much more relaxed than the autocorrelation time that characterizes the prediction ability of statistical schemes. In other words, we still can predict better than using conventional techniques [3].
In this paper we will adopt a formal definition of chaos, give examples of simple chaotic dynamical systems, sketch methods to predict the evolution of a chaotic system and apply chaotic systems to cryptography and signal prediction.