Autonomous Exploration and Control of Chaotic Systems

Elizabeth Bradley

Control algorithms that exploit chaotic behavior can vastly improve the performance of many practical and useful systems. Phase-locked loops, for example, are normally designed using linearization. The approximations thus introduced lead to lock and capture range limits. Design techniques that are equipped to exploit the real nonlinear nature of the device loosen these limitations. The program Perfect Moment is built around a collection of such techniques. Given a differential equation and two points in the system’s state space, it automatically selects and maps the region of interest, chooses a set of trajectory segments from the maps, uses them to construct a composite path between the points, and causes the system to follow that path via appropriate parameter changes at the segment junctions. Rules embodying theorems and definitions from nonlinear dynamics are used to limit complexity by focusing the mapping and search on the areas of interest. Even so, these processes are computationally intensive. However, the sensitivity of a chaotic system’s state-space topology to the parameters of its equations and the sensitivity of the paths of its trajectories to the initial conditions make this approach rewarding in spite of its computational demands. Controlled trajectories found by this program exhibit a variety of interesting and useful properties. For example, detours through chaotic regions can be used to steer trajectories across boundaries of basins of attraction, effectively altering both the geometry of and convergence properties within a particular convergence region -- such as the capture range of a phase-locked loop circuit.

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