9704639 Bollt This project addresses problems in controlling chaos and controlling symbol dynamics with small perturbations. Recent work has demonstrated the utilization of symbolic dynamics representation of controlled chaotic orbits for communications with chaotic signal generators. The evolution of trajectories of a chaotic dynamical system is equivalent to symbolic dynamics in an appropriate symbol space. The recent advent of controlling chaos using the OGY (Ott, Grebogi, and Yorke) technique and a variety of targeting algorithms, with physical applications, has demonstrated that chaos can be mastered, and inherent instabilities can be used as an advantage in allowing small deliberate perturbations to cause large signal variations. Coupling control of chaos through small perturbations with learning the grammar of the corresponding symbol dynamics means that the control perturbations are actually a coding scheme on the original dynamics. Controlling symbol dynamics using a map based description of the dynamics requires resolution of the following issues: learning the response of map iterates to variations in the control parameters, learning the semi- conjugacy, or coding function, between the dynamics of the map on the attractor and the grammar of the corresponding symbol dynamics, and finding the minimal grammar which is dependent on the appropriate choice of the partition in phase space. In particular, extensions include on-going work on communicating in higher dimensional dynamical systems. Investigations include issues of practical noisy environment control versus bandwidth trade-off and issues of ergodicity, learning system response to controls, and symbol dynamics of a chaotic dynamical system which is known only by time-series embedding of experimental data, and the further development of practical grammar learning algorithms. This new field of nonlinear communication theory promises to develop into a new paradigm offe ring a general information transmission technique, useful to both electronic and optical media, which should find wide applications in civil and military communications infrastructures. A great deal of recent research in applied and theoretical dynamical systems has been focused on taking advantage of the fact that a chaotic dynamical system can be controlled. The sensitive dependence characteristic of chaos is actually advantageous to building a highly agile control system in which a small deliberate system variations can cause a large response; the so called ``butterfly effect" allows us to steer the system responses with extremely small powered controls. Ergodic theory tells us that a chaotic system can be considered as an unlimited information source; and control of chaos allows us to manipulate this information flow with extremely low-powered controls. An example application is a high-powered signal generator (e.g. an electronic circuit), which operates intentionally in the chaotic regime, so that a small-scaled piggy-back controller circuit, on the micro-chip scale, has the ability to accurately manipulate high-powered message bearing signals. This method is in contrast to standard linear communication techniques, in which a high powered electronic circuit requires an equally large-scaled switching device to affect the large power variations required to transmit a high powered message. Not only is communicating through control of chaos applicable to one-dimensional dynamics, as previously demonstrated, but also applies to the more widely typical class of higher dimensional chaotic dynamics found in nature. Communicating by control of chaos promises numerous practical applications including the engineering of new and simple electronic communications devices, and new and simple optical communications devices, as well as the modeling of phenomenon in biology, chemistry, and cognitive science.
