I am trying to find a numerical solution for a system of nonlinear equations involving forced oscillations.

Below are the simplified equations I am trying to solve and the results from NDSolve with certain initial values. Note that some more terms and coefficients may be added in practice.

As you can see from these results, the system is known to reach a steady state after a certain time constant or so.

I am now mainly interested in the steady state rather than the transient state.

Usually, setting the NDSolve time setting sufficiently large and extracting the solution for the last cycle is sufficient. On the other hand, there are cases where the time constant of the system is very long relative to the period of the forced oscillation, and in this case, finding the steady solution can be very time consuming.

In such cases, I am looking for a way to find the steady-state solution in a short time.

As a simple idea, I guessed that I could find a stationary solution by changing the boundary conditions of NDSolve to periodic. However, when I try to do that with the equations I want to solve, I get an error output.

Due to the nonlinearity of the system, approaches using Laplace or Fourier transforms are also difficult.

Are there any good methods for such differential equations?

Sorry if there is something wrong with my English.

Regars,