= Hamiltonian blow-up of a reflection-symmetric double-zero point
{title2=$H=u^2+sv^2/2-v^4/8$}
Near a nilpotent origin with $\sigma=1$, $\mu=0$, the weighted scaling $u_{\rm old}=\varepsilon^2u$, $v_{\rm old}=\varepsilon v$, $\tau=\varepsilon t$, $\mu_{\rm old}=\varepsilon^2\mu$, $\sigma=1+\varepsilon^2s$ makes the leading cubic confinement flow $u'=-sv+v^3/2$, $v'=2u$. Direct differentiation proves conservation of $H=u^2+sv^2/2-v^4/8$. For $s>0$ the origin is a centre and $(0,\pm\sqrt{2s})$ are saddles. Their heteroclinic connections have $H=s^2/2$ and $u=\pm(2s-v^2)/(2\sqrt2)$ for $|v|\le\sqrt{2s}$. Positive lower energy levels surround the centre; higher levels escape the potential well.
Back to article page
