= Hamiltonian limit of three-to-one forcing
{c}
{title2=$H=x^2+y^2+2x^2y-\frac23y^3$}
For small positive forcing-frame frequency $\omega$, use $\mu=\widehat\mu\omega^2$, $C=\omega z$ and $\tau=\omega T$ in the <three-to-one spatially forced amplitude equation>. The leading system is $z_\tau+iz=\overline z^{\,2}$. Writing $z=x+iy$ gives the <Hamiltonian system> $(x_\tau,y_\tau)=(H_y/2,-H_x/2)$ and the displayed <first integral>. The origin is a <center equilibrium>; three <saddle equilibria> at $(0,1)$ and $(\pm\sqrt3/2,-1/2)$ lie on $H=1/3$. The factorization $H-1/3=(1+2y)[x^2-(y-1)^2/3]$ reveals a triangular <heteroclinic cycle>, containing closed <periodic orbits> for every $0<H<1/3$.
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