A Time-Domain Minimum Residual Method for Solving Time-Delayed Nonlinear Systems

Studying a semi-analytical solution for time-delay coupled nonlinear systems contributes to the advancement of the nonlinear dynamics theory. In this paper, based on the Time-domain Minimum Residual Method (TMRM), we introduced a time-delay term and obtained the semi-analytical solutions for two types of time-delay nonlinear systems. First, the periodic solutions of the time-delay coupled van der Pol–Duffing system were solved in three different cases based on different values of the time delay. The results were compared with numerical solutions and the Multifrequency Homotopy Analysis Method (MFHAM), showing good consistency among the three methods. Second, the quasi-periodic solutions of the time-delay coupled van der Pol–Duffing system were similarly solved in three different cases and compared with the numerical solutions, showing excellent agreement as well. Finally, the slow–fast analysis method was used to analyze the equilibrium points and bifurcation points of the time-delayed nonlinear oscillations of magnetic levitation, and the periodic bursting solutions were solved analytically, which demonstrates strong consistency with the numerical results. This shows that time delay has a significant impact on the behavior and performance of the system. Introducing a time-delay term based on the TMRM is highly valuable for obtaining the analytical solutions of time-delay nonlinear systems.