Symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator

In this paper, complex symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator are predicted through an implicit discrete map. The implicit discrete map is obtained by the discretization of a second-order differential equation of the time-delayed Duffing oscillator. Using the theory of nonlinear discrete systems, period-1 motions in the time-delayed Duffing oscillator are determined analytically from the fixed points of the mapping structures of discrete nodes under the computational accuracy of $$10^{-10}$$ , and the corresponding stability and bifurcation of period-1 motions are determined by eigenvalue analysis. From the discrete nodes of period-1 motions, the finite discrete Fourier series is employed to determine the frequency–amplitude characteristics of period-1 motions in the time-delayed, Duffing oscillator. The stable and unstable, symmetric and asymmetric period-1 motions are presented, and the corresponding stability and bifurcations are illustrated clearly. Presented are the quantity levels of harmonic amplitudes which indicate the accuracy of the semi-analytical solutions of period-1 motions in such a time-delayed oscillator. From the analytical prediction, numerical simulations of complex period-1 motions in the time-delayed, hardening Duffing oscillator are completed. The amplitude spectrums of period-1 motions in the time-delayed Duffing oscillator are given, and the approximate, analytical expressions of period-1 motions can be obtained. For small excitation frequency, discrete time step size should be reduced to keep the same computational accuracy of discrete nodes. This method can be applied to the time-varying time-delayed nonlinear dynamical systems and other nonlinear dynamical systems.