Improving the predictions of the circle criterion by combining quadratic forms

By using a Lyapunov function which consists of different quadratic forms in various sectors of the (
u, (du/d\tau)
) plane, the prediction of the circle criterion that the null solution of
(d^{2}u/d\tau^{2}) + 2(du/d\tau) + f(\tau, u, (du/d\tau))\cdotp u = 0
is asymptotically stable for
0 \leq \alpha < f(\cdotp) < \beta
, with
\beta = (\sqrt{alpha} + 2)^{2}
, is improved to
\beta = \[\{\frac{(\sqrt{\alpha} + 1)^{2} + 1 + \sqrt{(\sqrt{\alpha} + 1)^{4} + 2 (\sqrt{\alpha} + 1)^{2} + 5}}{2}\}^{\frac{1}{2}} + 1 \]^{2}
.