If you want to study the stability of the equilibrium point it is easier directly study the direction of in .
Fernando Revilla
x'=-y-x^3
y'=x
Using linearization, I see that the only fixed pt, (0,0) is a center. But the real parts of the eigenvalues are 0, so maybe linearization hasn't worked. I tried changing to polar to analyze, and I now have
theta' = 1+rcos^3(theta)
r'=-r^3cos^4(theta)
What do these tell me about the vector field? Have I made a mistake?
If you want to study the stability of the equilibrium point it is easier directly study the direction of in .
Fernando Revilla
Thanks for answering.
The prompt to my question tells me to analyze using polar coordinates to see whether or not the origin is a center.
I think I did make a mistake in my first post. I now have
theta'=1+r^2cos^3(theta)sin(theta)
but I still don't know what that tells me.
Sorry the post is hard to read.
Right.
but I still don't know what that tells me.
I would insist that is better in this case to directly study de direction of the vector field. Draw and and you'll have partitioned:
...
This will provide you information about the existence of closed orbits around .
Fernando Revilla
I know that your way is easier, but my professor says I should be able to use the values I have of r' and theta' to determine the direction of the non-linear spiral by looking at the speed in the radial direction. I think the origin is a growing spiral because r' will be greater than 0. Does this make sense, or am I not seeing this correctly?