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Perturb the Hamiltonian blow-up of a reflection-symmetric double-zero point by $u^{\prime }=-sv+v^3/2+\epsilon (\mu -v^2/2)u+O({\epsilon }^2)$ and $v^{\prime }=2u+\epsilon (\mu -v^2/2)v+O({\epsilon }^2)$. Then $H^{\prime }=\epsilon (\mu -v^2/2)(2u^2+s{v}^2-v^4/2)+O({\epsilon }^2)$. On the positive heteroclinic branch $dt=dv/[\sqrt{2}(s-v^2/2)]$, so its first energy change is $\epsilon M/\sqrt{2}$, where $M={\int }_{-\sqrt{2s}}^{\sqrt{2s}}(\mu -v^2/2)(s+v^2/2)dv$. A simple zero gives cancellation of the leading stable/unstable manifold splitting and continues to a nearby heteroclinic bifurcation. Since $\partial M/\partial \mu >0$, there is a unique first-order balance for fixed $s>0$.

At $(\sigma ,\mu )=(1,0)$ the origin's matrix is $\left(\begin{array}{cc}0& 0\\ 2& 0\end{array}\right)$: it has a double zero eigenvalue and one eigenvector. Trace and determinant vary independently with $\mu$ and $\sigma$ there, so two parameters are needed. The symmetry of an odd function removes quadratic terms, giving a reflection-symmetric cubic double-zero normal form. The quadratic nonlinearity required for a generic Bogdanov–Takens bifurcation vanishes.

Use a prime for differentiation with respect to the rescaled time. Direct substitution, with all rescaled variables written without tildes, givesAt $\epsilon =0$, the equations are $u^{\prime }=-sv+v^3/2$, $v^{\prime }=2u$. Fordifferentiation gives $H^{\prime }=2u(-sv+v^3/2)+(sv-v^3/2)2u=0$. This is the Hamiltonian blow-up of a reflection-symmetric double-zero point.

For $s>0$, the equilibrium points are a centre equilibrium at $(0,0)$ and saddles at $(0,\pm \sqrt{2s})$. The centre equilibrium has eigenvalues $\pm i\sqrt{2s}$; at either saddle equilibrium they are $\pm 2\sqrt{s}$. Levels $0<H<s^2/2$ contain closed inner ovals around the centre equilibrium. The saddle equilibrium level hasThese two branches form a heteroclinic cycle connecting the saddles. The branch with $u>0$ runs from the lower saddle equilibrium to the upper one because $v^{\prime }=2u>0$; the other runs back. Outside the potential well, trajectories are open. The right panel of the preceding figure uses the coordinates $(v,u)$ to display the well and these directed connections.