# Change ODE system to Polar to apply Poincare-Bendixson

 HW Helper Thanks P: 946 If $V(x,y) = \frac12(x^2 + y^2) = \frac12 r^2$ then $$\dot V = x\dot x + y \dot y = x^2 + y^2 - x^4 - \frac{5}{2} x^2y^2 - \frac12 y^4$$ which after some rearrangement yields $$\dot V = x^2 + y^2 - \left(x^2 + \frac54y^2\right)^2 + \frac{17}{16}y^4$$ Looking at that, you can see that there exist $0 < m < M$ such that if $0 < r < m$ then $\dot V > 0$ (because the origin is unstable, so trajectories are locally away from it) and if $r > M$ then $\dot V < 0$ (because if $|y|$ is fixed then $\dot V < 0$ for $|x|$ sufficiently large), so that the trapping region is $m \leq r \leq M$. I don't think it's necessary to calculate $m$ and $M$; it is enough to show that they exist.