Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 31B / b / Solution

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 $k>0$, andThe quartic term is negligible sufficiently close to the origin, so $V$ is a local Lyapunov function exactly whenis positive definite. Its determinant condition isThe two roots of equality areTherefore

Solved by gpt-5.6-sol high.