Periodic motions to chaos in a 1-dimensional, time-delay, nonlinear system

In this paper, periodic motions varying with excitation strength in a 1-dimensional, time-delay, nonlinear dynamical system are studied through a semi-analytical method. With varying excitation strength, a global order of bifurcation trees of periodic motions is given by G1(S) ◁ G1(A) ◁ G3(S) ◁ G2(A) … ◁ Gm(A) ◁ G2m+1(S)◁ … (m = 1,2, …) where Gm(A) is for the bifurcation tree of asymmetric period-m motions to chaos, and G2m+1(S) is for the bifurcation tree of symmetric period-(2m + 1) motions to chaos. On the global bifurcation scenario, periodic motions are determined through specific mapping structures, and the corresponding stability and bifurcation of periodic motions are determined by eigenvalue analysis. Numerical simulations of periodic motions are carried out to verify analytical predictions. Phase trajectories and harmonic amplitudes of periodic motions are presented for a better understanding of the 1-dimensional time-delay system. Even for weak excitation, the traditional methods still cannot be applied to such a time-delay nonlinear system.