Homoclinic balance for a quadratic-force oscillator
= Homoclinic balance for a quadratic-force oscillator
{title2=$\lambda=\frac57\sqrt\kappa$}
For $u''=u^2-\kappa-\varepsilon(\lambda+u)u'$ with $\kappa>0$, the Hamiltonian homoclinic orbit has $H=2\kappa^{3/2}/3$. Its energy balance is $\lambda\int v^2dt+\int uv^2dt=0$. Along this orbit the ratio of the second integral to the first is $-5\sqrt\kappa/7$, giving the displayed simple zero and hence the leading homoclinic parameter curve.