= 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 yields
$$
\boxed{u'=-2uv+\varepsilon u(\lambda_1+v^2),\qquad
v'=\lambda_2+v^2+u^2,\qquad \theta'=1/\varepsilon,}
$$
where primes denote $d/d\tau$. In the limit $\varepsilon=0$, the <first integral> is $F=-u(\lambda_2+u^2/3+v^2)$, since $u'=F_v$, $v'=-F_u$.
Write $\lambda_2=-a^2<0$. The physical amplitude half-plane is $u\ge0$, 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 arc
$$
\boxed{u^2=3(a^2-v^2),\quad -a\le v\le a.}
$$
On the axis $v'=v^2-a^2<0$, giving a <heteroclinic orbit> from $(0,a)$ to $(0,-a)$. On the arc $v'=2(a^2-v^2)>0$, 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-59-fold-hopf-contours.png]
{title=Conserved amplitude contours and the heteroclinic cycle of the quadratic fold–Hopf system}
Back to article page
