Chaotic Dynamics in Nonlinear Feedback Systems: Final Report.

摘要

This report summarizes recent research on the dynamics of nonlinear feedback systems. We are primarily interested in phase portraits of such systems and the ways in which these portraits depend on parameters. Because of this, we are interested primarily in processes which have associated probability measures that are absolutely continuous with respect to Lebesgue measure. Such processes will have 'interesting' phase portraits. In this paper we note that the space of piecewise constant functions is invariant under the Perron-Frobenius operator associated with any piecewise affine function. We further show that this will imply the existence of a piecewise constant probability density which is invariant for a given piecewise affine mapping satisfying certain easily checked conditions. We address the 'inverse density problem:' given a density /rho/ defined on some region UCR/sup n/, is it possible to find a mapping F: U implies U which is ergodic on U and which admits /rho/ as an invariant density. We show that an ergodic interval map cascaded with a stable linear system will, under certain conditions, results in a system having a strange attractor. For one canonical such mapping, we calculate the fractal dimension of the resulting attractor as a function of the parameters in the linear system. 15 refs., 5 figs. (ERA citation 14:021576)