Past exam of the differential equations 1 course of the University of Oxford / 2023 / 2 / a / i

An equilibrium point $p=(x_0,y_0)$ of an autonomous planar system is a point where its vector field vanishes:For a locally well-posed ordinary differential equation, Lyapunov stability means that every prescribed neighbourhood can be protected by choosing a sufficiently small initial neighbourhood. More precisely, for every $\epsilon >0$ there exists $\delta >0$ such that every trajectory starting with $\Vert p(0)-p\Vert <\delta$ exists for all $t\ge 0$ and satisfiesThe number $\delta$ may depend on $\epsilon$ but not on the particular starting point. Asymptotic stability additionally requires convergence to the equilibrium point for all sufficiently close starting points.