How to construct nonlinear ODE systems with given condition?

[xsw001]

I have a general question about how to construct nonlinear ODE systems with given condition such as # of critical points with certain characteristics of the phase portrait of each critical point.

I have no problem solving any type of nonlinear ODE system. But to do the reverse order, I have hard time to find an appropriate nonlinear general system to start with.

If I find the right one, then I can proceed to find the critical points with all unknown constants. Then linearize them individually through Jacobian matrix. Use the given characteristics of the critical points based on the eigenvalues from Jacobian matrix to find the parameters of the unknown constants, then randomly choose the constant within the constraints to find a nonlinear system.

Any suggestions in general? Thanks.

 

[jvicens]

Constructing nonlinear ODE systems with specific conditions can be a challenging task, but there are some general guidelines that can help you in this process.

Firstly, it is important to have a clear understanding of the characteristics of the phase portrait that you want to achieve. This can include the number and type of critical points, as well as the stability and behavior of the trajectories around these points. Once you have a clear idea of what you want to achieve, you can start by considering simpler systems that have similar behaviors and then gradually increase the complexity to match your desired conditions.

In addition, it can be helpful to use known techniques and methods for constructing nonlinear ODE systems, such as the method of undetermined coefficients or the method of Lyapunov functions. These methods can provide a systematic approach to finding the parameters and constants that satisfy your desired conditions.

Furthermore, it is important to keep in mind that there may not be a unique solution to your problem. In some cases, there may be multiple nonlinear systems that can satisfy your conditions. In such cases, it is important to carefully choose the system that best represents the dynamics you are trying to model.

Overall, constructing nonlinear ODE systems with specific conditions can be a trial-and-error process, and it may require some experimentation and adjustments to find the right system. I hope these suggestions can help guide you in this process. Good luck with your research!