= Solution
For the <logistic differential equation>, separate variables to obtain $\log|y/(1-ay)|=rt+C$ away from the constant solutions. Applying the positive <initial condition> gives
$$
\boxed{y(t)=\frac{1}{a+(y_0^{-1}-a)e^{-rt}}=\frac{y_0e^{rt}}{1+ay_0(e^{rt}-1)},\quad t\ge0.}
$$
The formula also includes the constant solution $y_0=1/a$. Below $1/a$ positive solutions increase; above $1/a$ they decrease. They approach $1/a$ as $t\to\infty$. Equivalently, linearizing $F(y)=ry(1-ay)$ gives $F'(0)=r>0$ and $F'(1/a)=-r<0$. Thus \b[zero is unstable and $1/a$ is asymptotically stable.]
The <Forward Euler method> gives $y_{n+1}=(1+r\delta t)y_n-ra\delta t\,y_n^2$. Set $\lambda=1+r\delta t$ and $u_n=a(\lambda-1)y_n/\lambda$; multiplying the recurrence by this scaling factor gives the <logistic map>
$$
\boxed{u_{n+1}=\lambda u_n(1-u_n).}
$$
Its <fixed points> are $u=0$ and $u_*=1-1/\lambda$. The <fixed-point multipliers> are $F'(0)=\lambda$ and $F'(u_*)=2-\lambda$. Since $\lambda>1$, zero is always unstable. The positive <fixed point> is stable for $1<\lambda<3$ and unstable for $\lambda>3$. For $\lambda>3$, a small displacement obeys $\epsilon_{n+1}\simeq(2-\lambda)\epsilon_n$: its sign alternates and its amplitude grows, giving an oscillatory instability whose period is \b[$2\delta t$].
This is an artifact of <Euler discretization of logistic growth>. Exact small perturbations decay by the positive factor $e^{-r\delta t}$ per time step; a one-dimensional autonomous <differential equation> cannot have a nonconstant <periodic orbit>, since its solutions move monotonically between <equilibrium points>. The artificial instability occurs when $r\delta t>2$, beyond the stable step range of this discretization.
For $\lambda=3.2$, remove the <fixed point> factors from $F(F(u))-u$ to obtain $\lambda^2u^2-\lambda(\lambda+1)u+\lambda+1=0$. The resulting <logistic map two-cycle> is
$$
\boxed{u_-\simeq0.5130445,\quad u_+\simeq0.7994555,\qquad u_\pm=\frac{\lambda+1\pm\sqrt{(\lambda-3)(\lambda+1)}}{2\lambda}.}
$$
These points map to each other. Their <periodic-orbit multiplier> is $F'(u_-)F'(u_+)=4+2\lambda-\lambda^2=0.16$, so the two-cycle attracts nearby iterates. The cobweb shows a small initial departure from $u_*=0.6875$ growing with alternating sign and then settling into the finite rectangle between $u_-$ and $u_+$. The time trace shows the corresponding finite-amplitude oscillation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ia/paper-2-logistic.png]
{title=Logistic-map cobweb and alternating iterates approaching the stable two-cycle at parameter 3.2}
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