Periodic Flows in Time-delay Systems

Abstract

This chapter will present periodic flows in nonlinear time-delay dynamical systems through the discrete implicit mappings. The period-1 flows in nonlinear time-delay dynamical systems will be discussed first by the one-step discrete maps, and then the period-m flows in nonlinear time-delay dynamical systems will also be discussed through the one-step discrete maps. Multistep, implicit discrete maps will be used to discuss the period-1 and period-m motions in nonlinear time-delay dynamical systems. Two methods are presented herein. The first method is based on the time-delay discrete nodes interpolated by the neighbored two nontime-delayed nodes. The second method is based on the time-delayed nodes determined integration between two nontime-delayed nodes. This method is also called continuation method. Through the discrete nodes in periodic flows , the periodic flows will be approximated by the discrete Fourier series and the frequency responses of the periodic flows can be determined through amplitude spectrums.