Analytical Predictions of Period-1 motions to Chaos in a Periodically Driven Quadratic Nonlinear Oscillator with a Time-delay

In this paper, periodic motions in a periodically forced, damped, quadratic nonlinear oscillator with time-delayed displacement are analytically predicted through implicit discrete mappings of the corresponding differential equation. From mapping structures, bifurcation trees of periodic motions are achieved analytically, and the corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. From the analytical prediction, numerical results of periodic motions are illustrated to verify such an analytical prediction. The semianalytical method gives the analytical prediction of the periodic motions matching very well with the approximate analytical solution for the time-delayed, quadratic nonlinear system. The method can also be applied to other time-delayed nonlinear systems.