Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 59 / 4 / c / Solution

With the added term, $\dot{r}=r({\mu }_1-2x+x^2)$, while $\dot{\theta }=1$ and $\dot{x}={\mu }_2+x^2+r^2$. The prescribed rescaling yieldswhere primes denote $d/d\tau$. In the limit $\epsilon =0$, the first integral is $F=-u({\lambda }_2+u^2/3+v^2)$, since $u^{\prime }=F_v$, $v^{\prime }=-F_u$.

Write ${\lambda }_2=-a^2<0$. The physical amplitude half-plane is $u\ge 0$, with $v$ of either sign; it is the radial variable $u$, not the axial variable $v$, that is nonnegative. The centre equilibrium is $(u,v)=(a,0)$, at $F=2a^3/3$. The boundary saddle equilibria are $(0,\pm a)$, both at $F=0$. For $0<F<2a^3/3$, the contours are nested closed curves around the centre. The level $F=0$ consists of the boundary $u=0$ and the interior arcOn the axis $v^{\prime }=v^2-a^2<0$, giving a heteroclinic orbit from $(0,a)$ to $(0,-a)$. On the arc $v^{\prime }=2(a^2-v^2)>0$, giving the reverse heteroclinic orbit. Together they form a heteroclinic cycle. The other parts of the zero level and negative levels give nonclosed trajectories outside this bounded region. The requested contour sketch is shown in the physical radial half-plane.