= Solution
Scaling by $K$, the positive fixed point is $x=1$. Linearization $x_t=1+\epsilon_t$ gives $\epsilon_{t+1}=\epsilon_t-r\epsilon_{t-1}$, 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.
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