Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 31B (Dynamical Systems)

Set $y=\dot{x}$. Then $\dot{x}=y$, $\dot{y}=a+bx-y-x^2-y^2$.

Fixed points have $y=0$ and $x^2-bx-a=0$. They coalesce on $a=-b^2/4$, the saddle-node curve. Away from it, $x_{\pm }=(b\pm \sqrt{b^2+4a})/2$; the Jacobian has trace $-1$ and determinant $2x-b$, so $x_{+}$ is stable and $x_{-}$ a saddle. At $(a,b)=(0,0)$ the branches $x=0$ and $x=b$ cross in a transcritical bifurcation.

For $a=0$ and extended variable $\dot{b}=0$, the centre manifold has $y=bx-x^2+$ higher-order terms. Hence $\dot{x}=bx-x^2+\cdots$, the transcritical normal form: $x=0$ and $x=b$ exchange stability at $b=0$.

For $b=1$, set $A=a+1/4$ and $X=x-1/2$. The centre manifold has $y=A-X^2+\cdots$, so $\dot{X}=A-X^2+\cdots$. This is a saddle-node at $a=-1/4$, with two branches for $a>-1/4$.

For $a=-1,b=1$, $\dot{y}=-1+x-y-x^2-y^2\le -1/2<0$ after maximizing the two quadratic expressions. Thus no orbit is periodic. A proper forward bounding region or Lyapunov trapping function would, in a planar flow with no equilibria, force a periodic limit set by Poincare-Bendixson; therefore no such global bounding function exists.