Stability and Robust Control of Period Doubling Bifurcations

Abstract

This thesis addresses the stability and feedback control of period doubling bifurcations. We find formulas for determining the stability of nondegenerate and degenerate period doubling bifurcations. Specifically, we find a formula for the second- order coefficient and a formula for the fourth-order coefficient which assumes the second-order coefficient vanishes in the Taylor series expansion about the critical fixed-point associated with a period doubling bifurcation of the function which maps the amplitude of an emerging period-two point to the eigenvalue of the linearization of the system about the period-two point which determines the bifurcation's stability. We observe that the size of the expression for the first nonzero coefficient in the series expansion above increases rapidly as the coefficient's order and the system's size increase.We examine both static and robust feedback control of period doubling bifurcations. We find conditions under which linear feedback incorporating a washout filter can delay the parameter value at which a period doubling bifurcation occurs, and conditions under which a purely nonlinear washout filter-aided feedback which preserves the critical parameter value and the critical fixed-point of a period doubling bifurcation can adjust the stability of the bifurcation. We demonstrate that these robust feedback controls can suppress the chaotic behavior in systems which exhibit the period doubling route to chaos. Additionally, we show that if a discrete-time system is exponentially stabilizable by static linear feedback, then it is exponentially stabilizable by washout filter-aided linear feedback.