Periodic and Chaotic Motion of a Time-Delayed Nonlinear System Under Two Coexisting Families of Additive Resonances

Abstract

A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions (SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies (HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HBFs, two families of additive resonances can be induced in the forced response. The forced response under the additive resonance related to the HBFs can demonstrate periodic, quasi-periodic and chaotic motion. On the contrary, the forced response under the additive resonance associated with the shifted HBFs may exhibit period-three periodic motion and quasi-periodic motion. Bifurcation diagrams, time trajectories, frequency spectra, phase portraits and Poincaré sections are presented to show periodic, quasi-periodic, and chaotic motion of the time-delayed nonlinear system under the two families of additive resonances.