Solution
= Solution
For the modified system, $F=x^2+y^2$ satisfies
$$
\dot F=2F-4F^2=2F(1-2F).
$$
With $F(0)=1$, the logistic solution is
$$
\boxed{x(t)^2+y(t)^2
=\frac1{2-e^{-2t}}}.
$$
Consequently,
$$
\boxed{\lim_{t\to\infty}(x(t)^2+y(t)^2)=\frac12}.
$$