Bifurcation control of nonlinear systems with time-periodic coefficients

Abstract

A technique for the bifurcation control of nonlinear systems with periodic coefficients is presented. In such systems, bifurcations occur when one of the Floquet multipliers becomes +1 , -1, or a pair of complex multipliers reaches magnitude 1. The stability of the bifurcated periodic or quasi-periodic orbit is guaranteed by employing a nonlinear state-feedback control. First the Lyapunov-Floquet transformation is applied such that the linear part of system equations becomes time-invariant. Then through an application of the time-periodic center manifold reduction and time-dependent normal form theory one can obtain a completely time-invariant form of the nonlinear equation for codimension one bifurcations. The time-invariant normal form is suitable for the application of control strategies developed for autonomous systems. Then by transforming the results back to the original variables, one obtains the gains for the time-varying controller. The control strategy is illustrated through an example of a parametrically excited simple pendulum undergoing a symmetry breaking bifurcation.