 Solution

Scaling by $K$, the positive fixed point is $x=1$. Linearization $x_t=1+{ϵ}_t$ gives ${ϵ}_{t+1}={ϵ}_t-r{ϵ}_{t-1}$, with characteristic polynomial ${\lambda }^2-\lambda +r$. The Jury conditions put both roots inside the unit circle exactly for $0<r<1$. At $r=1$ the roots are $e^{\pm i\pi /3}$, sixth roots of unity. The discrete Hopf resonance therefore creates a small period-six branch; expansion of the nonlinear recurrence on the centre eigenspace supplies the nonzero amplitude.