= Solution
On the invariant <biological population> interval $0\leq x\leq1$, the <fixed points> are $0$ and $x_*=1-1/\mu$ when meaningful. The zero <fixed-point multiplier> is $\mu$: extinction attracts for $0\leq\mu<1$. At $\mu=1$, the nonzero branch crosses the origin in the full real-line family, giving a <transcritical bifurcation>. The nonzero point lies in the <biological population> interval for $\mu>1$ and has <fixed-point multiplier> $2-\mu$, so it is stable for $1<\mu<3$ and loses stability through $-1$ at $\mu=3$.
Factoring $F^2(x)-x$ and removing the <fixed points> gives the <two-cycle>
$$
\boxed{x_\pm=\frac{\mu+1\pm\sqrt{(\mu-3)(\mu+1)}}{2\mu},\qquad\mu>3.}
$$
Its two-step <fixed-point multiplier> is
$$
\boxed{F'(x_+)F'(x_-)=-\mu^2+2\mu+4.}
$$
It lies between $-1$ and $1$ exactly for $3<\mu<1+\sqrt6$. At the upper endpoint it crosses $-1$, creating the next period doubling. The <two-cycle> amplitude tends to zero like $\sqrt{\mu-3}$ at its birth, so the first flip is supercritical. At the equality endpoints linear <fixed-point multipliers> alone do not establish <asymptotic stability>.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2007-logistic-bifurcations.png]
{title=Logistic fixed points and the first stable two-cycle through the second period-doubling threshold}
<Period-doubling cascade> universality concerns the continuation of this <period-doubling cascade> in smooth unimodal maps with a quadratic maximum. After suitable rescalings their repeated two-step maps approach the same limiting shape; successive <parameter> spacings have the Feigenbaum ratio approximately $4.6692$ and the spatial rescaling has magnitude approximately $2.5029$. These are asymptotic statements involving later doublings, not numbers that can be established from the first two thresholds alone. The accumulation near $\mu\simeq3.56995$ is outside the first interval considered here. One should not claim <chaos> throughout the stable <two-cycle> interval.
Four useful dynamical properties to distinguish are \b[sensitivity to initial conditions], \b[<topological transitivity> or mixing], \b[<dense periodic points>], and \b[<aperiodic orbits> with persistent complex evolution]. They concern the <invariant set>, not an assertion that every orbit must be aperiodic. A positive <Lyapunov exponent> for typical initial data is an additional quantitative diagnostic.
At $\mu=4$, set $x_0=\sin^2\theta_0$. The double-angle identity gives the exact solution
$$
\boxed{x_n=\sin^2(2^n\theta_0),\qquad\theta_0=\arcsin\sqrt{x_0}.}
$$
For a direct topological proof use the <homeomorphism> $h(y)=\sin^2(\pi y/2)$ of $[0,1]$. Then $F_4\circ h=h\circ T$, where the full <tent map> is
$$
T(y)=\begin{cases}2y,&0\leq y\leq1/2,\\2(1-y),&1/2\leq y\leq1.\end{cases}
$$
Every branch doubles interval length until it crosses a fold; the finite binary cylinders have length $2^{-n}$ and map onto the whole interval after $n$ steps. Thus every open interval eventually has an iterate covering the full interval: the map is topologically mixing and sensitive. Repeating a finite left/right branch word gives a periodic itinerary; arbitrarily long prescribed prefixes locate such periodic points in arbitrarily small cylinders, proving density. Nonperiodic binary itineraries yield uncountably many <aperiodic orbits>. Conjugacy transfers these properties to $F_4$, proving chaotic behaviour while retaining its dense special <periodic orbits>.
As a metric check, its <invariant measure> has density $1/[\pi\sqrt{x(1-x)}]$ and its typical <Lyapunov exponent> is $\log2>0$. This is a typical-orbit assertion, not a claim about the <derivative> along every exceptional critical or preperiodic orbit. The explicit trigonometric solution and tent-map conjugacy explain why deterministic <iteration of a map> can nevertheless amplify uncertainty.
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