 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/31b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

A Lyapunov function for an equilibrium is a continuously differentiable function that is positive definite there and whose derivative along nonconstant trajectories is negative definite in a neighbourhood.

For

V = x^{2} + k y^{2}

, positivity requires

k > 0

, and

\dot{V} = - 2 a x^{2} - 2 ak y^{2} + 2 (3 - k) x y + 2 (x^{2} + y^{2}) (x^{2} + k y^{2}) .

The quartic term is negligible sufficiently close to the origin, so

V

is a local Lyapunov function exactly when

a x^{2} + ak y^{2} - (3 - k) x y

is positive definite. Its determinant condition is

4 a^{2} k > (3 - k)^{2} .

The two roots of equality are

k_{1, 2} (a) = 3 + 2 a^{2} \mp 2 a a^{2} + 3 = (a^{2} + 3 \mp a)^{2} .

Therefore

(a^{2} + 3 - a)^{2} < k < (a^{2} + 3 + a)^{2} .

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