I'm having MAJOR difficulty with a problem regarding stability of a DE. The problem goes as following:
Find the equation of the phase paths of x˙=1+x^2, y˙=−2xy. It is obvious from the phase diagram that y=0 is Poincaré stable. Show that for the path y=0, all paths which start in (x+1)^2+y^2=δ^2 subsequently remain in a circle of radius δ[1+(1+δ)^2] centered on y=0
Finding phase paths is easy, and is expressed by:
I now see the phase paths all converges to y=0, and therefor is poincarč stabile, but the last part of the exercise is causing me alot of problems...
To show that all paths which starts in (x+1)^2+y^2=δ^2, i have tried finding the tangent between the circle and the phase paths, which will give me the highest y-value at x=0. Since the phase paths are symetric around x=0, and maximun value at the point. But I only get complex solutions, and they are no good...
Any tips on how to go from here?