= Solution
The <trace> and <determinant> of the origin's <linearization> are $2\mu$ and $\mu^2+\sigma^2-1$. Thus its open region of linear <asymptotic stability> is
$$
\boxed{\mu<0,\qquad\mu^2+\sigma^2>1.}
$$
For $|\sigma|<1$ this means below the lower semicircle; for $|\sigma|\ge1$ all negative $\mu$ lie in the open stable region. The origin is a <saddle equilibrium> inside the <unit circle> and a source outside it with positive $\mu$.
For nonzero <equilibrium points>, put $\Delta=2-(\mu-\sigma)^2$. Existence requires $\Delta\ge0$ and $\mu+\sigma+\sqrt\Delta>0$. An equivalent region is
$$
\boxed{\{\mu^2+\sigma^2<1\}\ \cup\
\{\mu+\sigma>0,\ |\mu-\sigma|\le\sqrt2\}.}
$$
Inside the circle exactly one <radius> <polynomial root> is positive, giving two <equilibrium points>. Outside it, with $\mu+\sigma>0$ and $\Delta>0$, both <polynomial roots> are positive, giving four. On $\Delta=0$ with $\mu+\sigma>0$, the two radii coalesce in a <saddle-node bifurcation> at each antipodal point. On the circle a zero <radius> is the origin and is excluded from the nontrivial count. These distinctions are shown in the parameter sketch.
At $(\sigma,\mu)=(1/\sqrt2,-1/\sqrt2)$ the <linearization> has one simple zero and one negative <eigenvalue>, while the cubic <centre manifold> coefficient vanishes. The quintic coefficient is negative. It is a <codimension-two bifurcation> with a reflection-symmetric degenerate <pitchfork bifurcation>: a wedge of coexistence between an attracting origin and an attracting antipodal pair opens between a subcritical <pitchfork bifurcation> and the nearby <saddle-node bifurcation> curve. The intervening smaller-radius pair consists of saddles. Indeed at a nonzero <equilibrium point>, direct <differentiation> of the <polar coordinates> equations gives
$$
\operatorname{tr}J=2(\mu-S),\qquad\det J=2S(S-\mu-\sigma)=\pm2S\sqrt\Delta.
$$
The smaller-radius branch has negative <determinant>, while in the coexistence wedge the larger-radius branch has positive <determinant> and negative <trace>.
At critical boundaries the <linear stability analysis> test alone is inconclusive. The origin is nonlinearly attracting at a supercritical lower <pitchfork bifurcation>, at its stabilizing quintic endpoint, and at the supercritical <Hopf bifurcation> threshold, although decay is no longer exponential. It is unstable at the subcritical lower <pitchfork bifurcation>. The double-zero point $(\sigma,\mu)=(-1,0)$ is also attracting: the positive function $W=v^2+u^4/8$ has
$$
\dot W=-uv^3-v^2S-\tfrac14u^4S+\tfrac14u^3vS
\le-\tfrac12v^4-(\tfrac12-S/8)u^2v^2-\tfrac18u^4S<0
$$
near the origin away from zero. Here $|uv^3|\le(u^2v^2+v^4)/2$ and $|u^3v|\le(u^4+u^2v^2)/2$ prove the bound. The other double-zero point $(1,0)$ has an unstable quartic potential, as the blow-up below shows. The shading records the open <linear stability analysis> regions; these boundary remarks distinguish nonlinear from exponential stability.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-85-confinement.png]
{title=Origin stability, nonzero-equilibrium counts and the Hamiltonian separatrix of the cubic confinement model}
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