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

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.

Assume $X,Y$ are continuously differentiable near the equilibrium. Put $\xi =x-a$, $\eta =y-b$ and expand the vector field:This Jacobian matrix defines the linearized system. For distinct real eigenvalues, two negative values give a stable node, two positive values an unstable node, and opposite signs a saddle point of a dynamical system. A complex-conjugate pair with negative real part gives a stable focus, and one with positive real part an unstable focus. Thus a hyperbolic saddle has two real eigenvalues of opposite signs, equivalently $\det M<0$.

The linearization stability theorem transfers these stability conclusions to the nonlinear system when no eigenvalue has zero real part. Distinctness alone is insufficient: a zero eigenvalue or a purely imaginary conjugate pair leaves a nonhyperbolic equilibrium, for which the nonlinear terms must be examined. In the purely imaginary case the linearized trajectories form a centre, but that does not establish a nonlinear centre.

One useful form of the Bendixson-Dulac theorem is this: let $D\subseteq {\mathbb{R}}^2$ be open and simply connected, and let $X,Y,\psi$ be continuously differentiable on $D$. Ifhas one strict sign throughout $D$, then there is no nonconstant periodic orbit lying entirely in $D$. More generally the divergence may have a weak sign provided it is not identically zero on any nonempty open subset. The multiplier $\psi$ is a Dulac function. Green's theorem proves the criterion: the vector field is tangent to a hypothetical closed orbit, so its scaled flux across that boundary is zero, whereas the signed area integral of its divergence would be nonzero.