Limit cycle analysis of multivariable nonlinear dynamic systems

The analysis of limit cycle behavior in multivariable nonlinear systems and systems containing multiple nonlinear elements is examined. Characterization of limit cycle behavior about an isolated equilibrium is provided by the Hopf bifurcation theorem. Interpreting the Hopf bifurcation theorem in the frequency domain is discussed and a frequency-domain version of the Hopf bifurcation theorem is presented. The frequency-domain version of the Hopf bifurcation theorem provides a convenient method of establishing the existence and stability of a limit cycle in multivariable nonlinear systems. In conjunction with the study of two nonlinear dynamical systems, analytical techniques have been applied and enhanced to understand the behavior of limit cycles. Parametric instabilities in electric power networks associated with Hopf bifurcations are studied via the frequency domain version of the Hopf bifurcation theorem. The analysis of a flutter instability of a three machine, four bus electric power network has been completed by showing that the encountered instability is a subcritical Hopf bifurcation. The analysis emphasized the feasibility for application to systems of large scale and more complex models. The oscillatory behavior of a spacecraft attitude control system is investigated. It is shown that a commonly used, momentum exchange, spacecraft attitude control configuration gives rise to complex nonlinear behavior involving multiple limit cycles and strange attractors. The results are based on the analysis of a simple single axis problem.