Chapter 2 - Implicit resonances in time-delayed nonlinear systems

Abstract

Time delays, widely existing in many practical engineering systems, can induce novel dynamic behavior including the change of stability of the trivial equilibrium, occurrence of Hopf bifurcations, stable and unstable bifurcating periodic solutions, and the coexistence of bifurcating stable solutions in the time-delayed nonlinear systems without external excitations. In the presence of periodic excitations, the dynamic interaction between the stable bifurcating solutions and the periodic excitations can induce the resonances in the forced response of time-delayed nonlinear system in the neighborhood of Hopf bifurcations. This kind of resonances is established when the frequencies of the stable bifurcating solutions and the forcing frequency satisfy certain relationships and can only be discovered in seeking the approximate periodic solutions of the corresponding reduced ordinary differential nonlinear equations on the center manifold for Hopf bifurcations. In this regard, these resonances will be referred to here as implicit resonances in this chapter, which are used to distinguish from the resonances occurring in conventional nonlinear systems where the linear natural frequency needs to have certain relationships with the forcing frequency. To further exemplify implicit resonances, two coexisting families of three secondary resonances including superharmonic, subharmonic, and additive resonances are discussed for the forced response of a time-delayed quadratic nonlinear system subjected to the periodic excitation in the neighborhood of two-to-one resonant Hopf bifurcations. Here, the time-delayed quadratic nonlinear system without external excitations exhibits two coexisting bifurcating periodic solutions, which have different amplitudes and frequencies.