I am working on a nonlinear dynamics problem set and I have a question about deriving the equation that determines the x coordinates (x∗) of the fixed points of the model represented by these two ODEs.
$\dot{x} = y - x^3 + 3x^2 + I$
$\dot{y} = 1 - 5x^2 - y$
The fixed points are the intersections of the two ODEs, thus I think that you can obtain the nullclines and then set the equations for the nullclines equal to each other?
$\dot{x} -nullcline: y = x^3 - 3x^2 + I$
$\dot{y} -nullcline: y = 1 - 5x^2$
Making the equation to determine the values of x the simplified version of:
$x^3 - 3x^2 + I = 1 - 5x^2$
Is that the proper way to find the equation that determines the x coordinates (x∗) of the fixed points of the model represented by these two ODEs? I apologize if my question is silly!
Thanks so much.