 Solution

For $\lambda >0$ put $s=\sqrt{\lambda }$. The equilibrium points are $(s,0)$ and $(-s,0)$, withThe 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 $\mu \ne 0$; 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; thenSet $X=u$, $Y=-v/\omega$ 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 coefficientwhere $g=-X^2/\omega +XY$. The radial equation is $\dot{R}=(\mu -s)R/2+{\ell }_H{R}^3+\cdots$. Hence the Hopf bifurcation is 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.