Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 31A (Dynamical Systems)

The second equation gives $y=x(1-x)$. The first then says either $y=0$, giving the persistent equilibrium points $(0,0)$ and $(1,0)$, orgiving $x_{\pm }=((\beta +1)\pm \sqrt{(\beta +1{)}^2+4\mu })/2$ and $y_{\pm }=x_{\pm }(1-x_{\pm })$. These two points exist for $\mu \ge -(\beta +1{)}^2/4$.

At the origin the Jacobian matrix has trace $-1$ and determinant $-\mu$, so it is stable for $\mu <0$ and a saddle point for $\mu >0$. At $(1,0)$ the trace is $-1$ and determinant $\mu +\beta$, so it is stable for $\mu >-\beta$ and a saddle for $\mu <-\beta$. Stationary bifurcations occur atwith the first two coinciding when $\beta =1$.

At a non-axis fixed point the determinant and trace areThese formulas determine the solid stable and dashed unstable branches in the bifurcation diagram. The pair $x_{\pm }$ is born in a saddle-node bifurcation at $\mu =-(\beta +1{)}^2/4$. For $\beta \ne 1$, it exchanges branches with $x=1$ at $\mu =-\beta$ and with $x=0$ at $\mu =0$, both transcritical bifurcations. If $0<\beta <1$, the saddle-node lies to the left of $x=1$; if $\beta >1$, it lies to the right. For $\beta =1$, the saddle-node and the collision at $x=1$ combine into a pitchfork bifurcation at $\mu =-1$.

Put $x=1+u$, $y=v$, and $\mu =-1+\nu$, adjoining $\dot{\nu }=0$. ThenThe zero eigenvalue has centre direction $v=-u$. Solving the center manifold invariance equation givesSubstitution yields the reduced equationIts equilibria are $u=0$ and $u=\pm \sqrt{\nu }+O(\nu )$ for $\nu >0$, exactly matching the three branches meeting at the pitchfork in part (b).

For $\nu =\mu +1>0$ small, $(1,0)$ has two negative eigenvalues and is a stable node. The two nearby equilibria satisfy $u=\pm \sqrt{\nu }+O(\nu )$ and are saddles. Their stable and unstable separatrices form the local phase portrait; trajectories in the wedge between the stable manifolds approach $(1,0)$, while trajectories crossing an unstable direction leave the neighbourhood.