= Solution
For $\lambda>0$ put $s=\sqrt\lambda$. The <equilibrium points> are $(s,0)$ and $(-s,0)$, with
$$
J(x_*,0)=\begin{pmatrix}0&1\\2x_*&\mu+x_*\end{pmatrix},\quad
\det J=-2x_*,\quad\operatorname{tr}J=\mu+x_*.
$$
The positive branch is always a saddle. The negative branch is a sink for $\mu<s$ and a source for $\mu>s$; it is a node or focus according as $(\mu-s)^2-8s$ is positive or negative. At $\lambda=0$ the two branches meet in a <saddle-node bifurcation> when $\mu\ne0$; at $(\lambda,\mu)=(0,0)$ the double zero <eigenvalue> with a nontrivial Jordan block gives a <Bogdanov–Takens bifurcation>. There are no real <equilibrium points> for $\lambda<0$.
The <Hopf bifurcation> curve is $\mu=s>0$, equivalently $\lambda=\mu^2$ on its positive-$\mu$ branch. Its frequency is $\omega=\sqrt{2s}$. To determine the direction, shift $u=x+s$, $v=y$ at this threshold; then
$$
\dot u=v,\qquad \dot v=-\omega^2u+u^2+uv.
$$
Set $X=u$, $Y=-v/\omega$ to obtain $\dot X=-\omega Y$, $\dot Y=\omega X-X^2/\omega+XY$. A quadratic near-identity transformation, or harmonic averaging with its quadratic correction retained, gives the radial cubic coefficient
$$
\ell_H=\frac{-g_{XY}g_{XX}}{16\omega}
=\frac1{8\omega^2}>0,
$$
where $g=-X^2/\omega+XY$. The radial equation is $\dot R=(\mu-s)R/2+\ell_H R^3+\cdots$. Hence the <Hopf bifurcation> is \b[subcritical], with an unstable small <periodic orbit> on the stable-focus side $\mu<s$.
The saddle's outgoing separatrix can return to its incoming separatrix, creating a <homoclinic orbit>. This <global bifurcation> cannot be detected from the <equilibrium point>'s <Jacobian matrix> alone. The weakly Hamiltonian scaling below locates its curve and shows how the unstable Hopf cycle grows into the saddle loop. The final diagram includes the local saddle-node and Hopf curves and this global curve; it describes the neighborhood of the double-zero point.
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