Eur. Phys. J. Special Topics 228, 1747-1765 (2019)
https://doi.org/10.1140/epjst/e2019-800243-y

Regular Article

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

¹ Department of Mechanical Engineering, California Polytechnic State University, San Luis Obispo, CA 93407, USA
² Department of Mechanical and Industry Engineering, Southern Illinois University, Edwardsville, Edwardsville, IL 62026-1805, USA

^a e-mail: aluo@siue.edu

Received: 30 December 2018
Received in final form: 11 July 2019
Published online: 25 September 2019

Abstract

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

     G_1(S) ◁ G_1(A) ◁ G_3(S) ◁ G_{2(A) …} ◁ G_m(A) ◁ G_2m+1(S)◁ … (m = 1,2, …)

where G_m(A) is for the bifurcation tree of asymmetric period-m motions to chaos, and G_2m+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.