Guiding Chaotic Orbits

This report studies chaotic systems with particular emphasis on the recently developed method of E. Ott, C. Grebogi and J. A. Yorke [Ott 1990] (the OGY method) for controlling such systems. Concepts useful in understanding chaos in general are introduced. We can conceptualise chaotic systems as arising from classes of differential equations having particularly intractable solutions sets. However, in many applications the underlying equations are unknown, one works from observations of measurable parameters of the system. The use of successive samples of a single variable (or few variables) to generate an embedding with a view to reconstructing the details of an attractor for a higher dimensional dynamic system was suggested in [Packard 1980] and a frequently quoted embedding theorem [Takens 1980] establishes the existence of such models for homogeneous systems: if the underlying state space of a system has d-dimensions then the embedding space needs to have at most (2d + 1) dimensions to capture the dynamics of the system completely. These results were later generalised and improved by [Levin 1993]. It is a remarkable fact that much of the dynamics of a high dimensional system can be recovered from a suitable embedding of a single variable, but in practice a critical factor in the accuracy of such reconstructions is the sampling delay. In this report a number of existing techniques for deriving delay time, sampling delays, suitable for reconstruction are examined and improved methods for estimating jump time, the time between each embedding space vector, and embedding dimension are elaborated. Combining a new technique (the Γ-test) [Stefánsson 1995] with existing techniques leads to a new and effective automated framework for attractor reconstruction. Choice of the jump time is critical in extracting an infinite number of nearly periodic behaviours which exist within a reconstructed chaotic attractor. Study of techniques in choosing the jump time led us to discover the creep phenomenon, where successive embedding space vectors remain nearby and slowly cover the entire attractor in sections. Techniques which facilitate attractor reconstruction become critical when one seeks to apply the OGY method in cases where the mathematical equations which define the dynamic system are not available. The original OGY method and a variation due to U. Dressler and G. Nitsche [Dressler 1992] are consolidated into a single formal framework and comparative results are presented. One major disadvantage of the OGY method is an inability to control complex system behaviour. The method is extended to control complex behaviours which exists within the system. The ability to control chaotic systems may possibly help us to understand some aspects of biological brain function. It has been suggested by W. J. Freeman [Freeman 1991] that chaos is evident in the brain and may play an important role in cognitive processes. According to this model the brain at rest is in a chaotic I mode. Upon receiving a sensory input the response to the stimulus is more ordered, more nearly periodic during perception, than at rest. In the language of nonlinear dynamics this may be interpreted as a shift from a chaotic orbit to a periodic orbit. Thus there are possibilities that ideas based on the OGY method could be used to simulate this aspect of brain function.