Steady state bifurcation of a periodically excited system under delayed feedback controls

摘要

This paper investigates the steady state bifurcation of a periodically excited system subject to time-delayed feedback controls by the combined method of residue harmonic balance and polynomial homotopy continuation. Three kinds of delayed feedback controls are con sidered to examine the effects of different delayed feedback controls and delay time on the steady state response. By means of polynomial homotopy continuation, all the possible steady state solutions corresponding the third-order superharmonic and second-subhar monic responses are derived analytically, i.e. without numerical integration. It is found that the delayed feedback changes the bifurcating curves qualitatively and possibly eliminates the saddle-node bifurcation during resonant. The delayed position-velocity coupling and the delayed velocity feedback controls can destabilize the steady state responses. Coexis ting periodic solutions, period-doubling bifurcation and even chaos are found in these con trol systems. The neighborhood of the periodic solutions is verified numerically in the phase portraits. The various effects of time delay on the steady state response are investi gated. Many new phenomena are observed.