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 , 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} .