Solving nonlinear ODE with ODE45 fails at bifurcation point?

[JasonMech]

Hello everybody,

Background and problem description

I have derived an analytical expression for an implicit frequency response function. To verify it, I would like to check with a numerical solution. For very weak nonlinearities, congruence is obtained. For weak nonlinearities, the numerical solution cannot follow the solution curve (a hysteresis curve). It seems very much like the solution curve jumps at the lower bifurcation point. I have attached a screenshot of a graph generated in Matlab showing how the numerical solution behaves alongside the analytical solution.

Attempts to solve the problem

I have tried to integrate from both [0 n] (where n is an arbitrary time constant) and from [-n 0]. I have also tried to change the ODE solver [ODE45, ODE23s, ODE23t, ODE23tb and ODE15s (in case the problem is stiff)]. I update the initial guess provided to the ODE-solver for each iteration. Changing the tolerances when using the different ODE solvers has not provided any different results. I have considered implementing other approaches such as the tangent method and/or the pseudo-arclength continuation method. But I am not sure whether these approaches will solve my problem.

Any help/inspiration/ideas/thoughts/tricks/... would be highly appreciated!

Kind regards,
Jason

 

[jvicens]

Hello Jason,

I can understand your frustration with trying to verify your analytical expression with a numerical solution. It seems that you have already tried various approaches and different ODE solvers, but have not been able to get congruence between the two solutions.

One suggestion I have is to check your initial conditions and make sure they are consistent between the analytical and numerical solutions. Sometimes small differences in initial conditions can lead to significant differences in the solution curves.

Another approach you could try is to use a different software or programming language to solve the ODE. Sometimes different solvers or algorithms can produce different results, so it may be worth trying a different tool.

Additionally, the tangent method and pseudo-arclength continuation method could potentially be useful in solving your problem. These methods are commonly used in nonlinear dynamics and bifurcation analysis, and may provide more accurate results compared to the ODE solvers you have already tried.

I hope these suggestions are helpful and wish you the best of luck in finding a solution to your problem. Don't hesitate to reach out if you need further assistance or have any other questions.