Determine whether the following nonlinear system has periodic solutions.
\begin{align} x'&=x^2+y^2+1\\ y'&=x^2-y^2 \end{align}
I have tried using polar coordinates but it just becomes complicated and doesn't give any information. I've also tried the candidate Lyapunov function
$$V(x,y)=\frac{1}{2}(x^2+y^2)$$
but, since $\dot{V}$ is not negative definite, I can't apply Poincaré-Bendixson theorem since I can't find a positively invariant set.
I have also tried to show that there can be no orbits by using Bendixson's criterion, but the divergence is not $\neq 0$ thus the result is inconclusive. Any idea how to approach this?
As highlighted in the previous comment, since $1\leq\dot{x}(t)$ it comes out easily that the orbits cannot be closed and hence be periodic, otherwise there would be a certain time interval where the x decreases.