- #1

Zebx

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Hi. I'm not sure about something related to the equilibrium points (or fixed points) of a non linear ode system solution. As far as I know, to check if an equilibrium point exists, I need to put the function of my ode system equal to zero. Then once the point is found, I can use it to evaluate its stability. If I have 2 equilibrium points for example, one of which is linearly stable, then it means that if I use this point as my initial condition I will get a constant solution. Moreover, if I evaluate the Jacobian of my system in the stable equilibrium point, I can use the eigenvalues to check what timesteps I have not to use in order to have stable solutions.

I was trying to apply what said before (supposing it is all correct) to my ode system. I already checked the methods used works and plot some solution, but I wanted to try to see if I could apply the above analysis properly. For instance I started by considering a 2-body problem with a star at the center of the system and a planet orbiting around it. Looking at the equations I already have a problem: the function is zero just if I use ##(x=0, y=0, z=0)##, but since I have ##\sqrt{x^2 + y^2 + z^2}## at the denominator I would get infinites. So I thought that maybe it's normal cause, if I actually had an equilibrium point for my system and I used it as starting condition, I should get a constant solution, but it's impossible to have a constant solution for my problem since I have a body moving in a star potential. So basing on this I thought that maybe this is one of these cases in which you can't have a clue about stability from jacobian study.

Is my reasoning correct?

I was trying to apply what said before (supposing it is all correct) to my ode system. I already checked the methods used works and plot some solution, but I wanted to try to see if I could apply the above analysis properly. For instance I started by considering a 2-body problem with a star at the center of the system and a planet orbiting around it. Looking at the equations I already have a problem: the function is zero just if I use ##(x=0, y=0, z=0)##, but since I have ##\sqrt{x^2 + y^2 + z^2}## at the denominator I would get infinites. So I thought that maybe it's normal cause, if I actually had an equilibrium point for my system and I used it as starting condition, I should get a constant solution, but it's impossible to have a constant solution for my problem since I have a body moving in a star potential. So basing on this I thought that maybe this is one of these cases in which you can't have a clue about stability from jacobian study.

Is my reasoning correct?

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