Hello,

It sounds like you're trying to compute the frequency-amplitude response curve for a system and are having difficulty using the arc length continuation method, despite verifying the correctness of your equations.

Here are a few suggestions to troubleshoot and potentially resolve the issue:

1. Check Initial Guesses: Both `fsolve` and `vpasolve` rely on good initial guesses. Ensure that the initial conditions you're providing are close to the expected solution. Arc length continuation is sensitive to this, especially for nonlinear problems where multiple solutions may exist.

2. Scaling of Parameters: Make sure that your parameters are properly scaled. Sometimes, nonlinear equations can become ill-conditioned if the variables or parameters span different magnitudes, leading to numerical instabilities.

3. Arc Length Continuation Settings: The arc length continuation method requires setting an appropriate step size for continuation. If the step size is too large, the method may skip over important solution branches or lose stability. Try adjusting the step size to see if it improves convergence.

4. Bifurcation Points: If you're working near a bifurcation point, the method may struggle. It's worth checking if the continuation algorithm is correctly handling turning points or bifurcations.

5. Numerical Tolerance: Ensure that the solver's tolerance settings are tight enough for accurate solutions. Loosely defined tolerances might result in convergence issues.

If these suggestions don't help, feel free to share more details about the specific problem you're encountering. I'd be happy to take a closer look.