Past exam of the differential equations 1 course of the University of Oxford / 2017 / 1 / a / i

The autonomous planar system has two real state variables, so its phase space is a subset of ${\mathbb{R}}^2$. It is autonomous because its vector field $(X(x,y),Y(x,y))$ has no explicit time dependence. The nullclines are $X(x,y)=0$ and $Y(x,y)=0$, on which the respective coordinate's instantaneous rate of change vanishes. A critical point of a dynamical system is an intersection where $X=Y=0$; the constant trajectory at that point is an equilibrium of an autonomous differential equation. A nullcline alone need not consist of equilibria.