Which ode should I choose for 2 sets of differential equations?

Hi all

I have a set of 2 differential equations ( regardless of what they are or what they do) I just want to discuss the possibility to improve the solving in mathematical point of view

the expected solution is the blue line. while the red one is what I do using ode45.

the convergence rate is not very good. I also tried with different initial conditions, but it seems like with a bigger r , I get worse convergence ,that it takes 5 minutes for the blue curve bellow, and 6 hours for the yellow one. so not very convenient to solve. bellow is just an example of the trials

[Steven Lord]

Can you show the code you wrote to solve this using ode45 as well as state the expected answer and how you obtained that answer? That may help us understand why the results are different after about U = 5.

[farzad]

Steven, immagine that you are only given the formulas, how would you solve it and at least, how would you evaluate it ?

[farzad]

So any ideas ?