Purpose <p>In this paper, we focus on solving the periodic solution, quasi periodic solution and periodic bursting solution of the time delay coupled van der Pol-Rayleigh system with parameter excitation.</p> Methods <p>This paper mainly uses the time domain minimum residual method(TMRM). Firstly, the periodic, quasi periodic and periodic bursting solutions for the non-time delayed case are determined using the TMRM. Secondly, the extended time domain minimum residual method, developed by incorporating a time delay term, is applied to determine the periodic, quasi periodic, and periodic bursting solutions of the time delay system. Finally, The obtained approximate analytical expressions are compared with numerical solutions obtained using the Runge-Kutta (RK) method to validate their accuracy.</p> Results <p>For both the case where the time delay is 0 and the case where the time delay is not 0, we obtain analytical expressions for the periodic solution, periodic bursting solution and quasi periodic solution of this system. The comparison with the numerical value shows that the solution is in good agreement, which indirectly proves that this method is also effective for time delay systems.</p> Conclusions <p>Time delay systems have a wide range of application background, and the solution of such nonlinear systems has been a hot research topic. In this paper, the analytical solution of the system is obtained by using the time domain minimum residual method. And the comparison with the numerical solution shows that the time domain minimum residual method is an effective method for solving time delay systems. The research in this paper provides a reference for solving time delay systems.</p>