Chaotic Phenomena and Their Engineering Relevance

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The objective of research is to understand the behavior of nonlinear dynamical systems using theory and extensive numerical simulation. Prototype models including second-order oscillators have been selected to include some previously neglected but essential properties of real physical and engineering systems. Basic insight is gained by avoided unnecessary camplications.A comprehensive review and classification of hifurcations involving one parameter was completed. Safe, explosive and dangerous types were distinguished, and related to phenomena of fundamental concern in applications : continuity or discontinuity of responses, hysteresis, intermittency and indeterminate outcomes. Study of escape from potential wells was extended to a wide range of volues of the damping coefficient. The close connection between optimal escape and resonance was confirmed ; subtle but important changes in bifurcation patterns were discovered, and their significance for experimental studies was clarified.We have considered the nonlinear dynamical system with time delay described by the Minorsky equation (see Minorsky, J.Appl.Phys, Vol.19, 1948), which he introduced during his studies of active ship stabilization. Differential equations with a time delay, sometimes called differential-difference equations, have an infinite dimensional phase space. The global features of the basin boundaries are not easily grasped, but they have great practical importance.In earlier work, we made some progress by making 'carpet bombing' experiments from a grid of starts in one cross-section of the phase space : but now we have focused attention on the unstable basic sets governing the basin boundaries, which we have located numerically using the straddle orbit technique. This approach gives much more insight into basin structure than carpet bombing ; for example, a stability index involving distance from attracter to unstable basic set may be computed. This will be pursued in future research.