Classify the fixed point at the origin of a dynamical system.

Question

If we have a system $\dot x = -y+ax^3$ and $\dot y = x+ay^3$ I need to classify the fixed point at the origin for all real values of a.

So I know we have to make the change of variables $ x = r\cos\theta$ and $y = r\sin\theta$ and $x\dot x+y\dot y = r\dot r$

So substituting this is gives, $r\dot r = a(x^4+y^4) $ and $ \dot \theta = $ something that isnt very nice.

Please help me finish this problem. Thanks a lot.

Answer 1

"Substituting" gives $\dot r=ar^3(\cos^4\theta+\sin^4\theta)$. Note that $\cos^4\alpha+\sin^4\alpha\geqslant\frac12$ for every $\alpha$. Hence:

• if $a\gt0$ then $\dot r\geqslant\frac12ar^3$ hence $r\to\infty$ for every $r(0)\ne0$ (unstable)
• if $a\lt0$ then $\dot r\leqslant-\frac12|a|r^3$ hence $r\to0$ (stable)
• if $a=0$ then $r=r(0)$ (center)

(When $a\gt0$, the solution blows up in finite time in the sense that, for every $r(0)\gt0$, $r(t)\to\infty$ when $t\to t_*$ for some finite $t_*$ depending on $(r(0),\theta(0))$ and such that $\frac12\leqslant ar(0)^2t_*\leqslant1$. When $a\leqslant0$, the solutions are defined at every positive times.)