To show that $\dot{x} = -y - x +y^2 - x^2,\; \dot{y} = xy$ has no periodic orbits.

Question

I've tried using index theory, but there's a non-isolated fixed point at $(0, 0)$ (the remaining fixed points at $(0, 1)$ and $(-1, 0)$ are saddles).

The terms in the equations have even indices and the wrong signs, so using a Liapunov function $V(x, y) = x^n + a y^m$ won't work.

I've tried using the Dulac function $g = 1/xy$ to get that $\nabla \dot (gF) = \frac{1 - y}{x^2} - \frac{1}{y} < 0$ for $y > 1$, which isn't a strong enough result.

Answer 1

A phase portrait shows there can be no periodic orbits. Each trajectory must either approach a fixed point or go off to infinity. For a detailed proof of this, you could examine the fate of solutions starting out in each of the regions into which the isoclines divide the plane.