Dynamical systems and chaos

MotivationH

The long run

In 1961 the meteorologist Edward Lorenz restarted a weather simulation halfway through, typing in a value printed to three decimal places, $0.506$ , instead of the $0.506127$ stored in the machine. He went for a coffee. When he came back, the new forecast had drifted completely away from the old one. The equations were deterministic and the change was one part in ten thousand. Lorenz had found sensitive dependence on initial conditions, and with it the reason why weather cannot be forecast more than a couple of weeks ahead, however good our computers become.

Fifteen years later the ecologist Robert May studied the simplest model of an insect population with limited food: next year's population is $x_{n+1} = r x_n(1 - x_n)$ . For small growth rates $r$ the population settles to a steady value. Raise $r$ past $3$ and it alternates between two values, then four, then eight. Past $r \approx 3.57$ it behaves erratically, never repeating — from a quadratic formula that a child could iterate on a calculator.

Most differential equations cannot be solved by a formula. Henri Poincaré, studying the motion of three bodies under gravity in the 1880s, changed the question from "what is the solution?" to "what does it do in the long run?": does it settle down, repeat, or wander? That qualitative theory is the subject of this lesson. It covers fixed points and their stability, cycles and the theorem of Poincaré and Bendixson that finds them, bifurcations where the behaviour changes, and finally chaos, with an honest account of what computers can and cannot tell us about it.

IntuitionH

Valleys, spirals, stretching and folding

A dynamical system is a rule for how a state changes. It may change in discrete steps, $x_{n+1} = f(x_n)$ (a map), or continuously, $\dot x = F(x)$ (a flow). The states form the phase space, and the history of one state is its orbit. A pendulum's state is its angle and angular velocity, a point in a plane. Its orbits are closed curves (swinging), wavy lines (spinning over the top), and the two equilibria (hanging down, balanced upright).

Fixed points are states that never change. Some are stable, like a ball at the bottom of a valley: nudge it and it returns. Others are unstable, like a ball on a hilltop. For a map of the line the test is the derivative. Near a fixed point $x^*$ , $f(x) - x^* \approx f'(x^*)(x - x^*)$ , so distances to $x^*$ are multiplied by about $|f'(x^*)|$ at each step. They shrink if $|f'(x^*)| < 1$ and grow if $|f'(x^*)| > 1$ .

Cycles are orbits that repeat. In the plane, a flow can wind towards a closed loop — a limit cycle — like the steady beating of a heart or the squeal of a violin string. In the plane that is about the most complicated thing a bounded orbit can do, because orbits cannot cross.

Chaos needs room. Its mechanism is stretching and folding. The logistic map at $r = 4$ stretches $[0, \frac12]$ onto $[0, 1]$ and folds $[\frac12, 1]$ back onto $[0, 1]$ , like a baker kneading dough. Two nearby points are pulled apart at every step until they end up on opposite sides of a fold, after which their futures are unrelated.

Cobweb diagram for a map $f$: from $x_0$ go vertically to the graph, horizontally to the diagonal $y = x$, and repeat. Fixed points are where the graph meets the diagonal. Here the slope at the fixed point is between $-1$ and $0$, and the orbit spirals inwards.

DefinitionsH

Definitions

Definition— map, flow, orbit

A (discrete) dynamical system is a map $f : X \to X$ on a metric space $X$ . The orbit of $x$ is $x, f(x), f^2(x), \dots$ , where $f^n$ is the $n$ -fold composite. A flow on $\R^d$ is given by an autonomous ODE $\dot x = F(x)$ with $F$ of class $C^1$ . By the Picard–Lindelöf theorem through each point passes a unique solution $t \mapsto \phi_t(x)$ , and $\phi_{s+t} = \phi_s \circ \phi_t$ wherever defined.

Definition— fixed and periodic points, stability

$x^*$ is a fixed point of $f$ if $f(x^*) = x^*$ , and an equilibrium of a flow if $F(x^*) = 0$ . $x$ is periodic with period $n$ if $f^n(x) = x$ ; its minimal period is the least such $n \ge 1$ . A fixed point is attracting if every orbit starting close enough converges to it. It is Lyapunov stable if orbits starting close enough stay close for all future time, and asymptotically stable if it is both. It is repelling if some neighbourhood is left eventually by every orbit starting in it, except $x^*$ itself.

Definition— hyperbolic fixed point

A fixed point $x^*$ of a $C^1$ map is hyperbolic if no eigenvalue of $Df(x^*)$ has absolute value $1$ . An equilibrium of a flow is hyperbolic if no eigenvalue of $DF(x^*)$ has real part $0$ . For a periodic orbit of period $n$ of a map of $\R$ , its multiplier is $(f^n)'(x) = f'(x)f'(f(x))\cdots f'(f^{n-1}(x))$ , by the chain rule.

Definition— limit cycle, ω-limit set

A periodic orbit of a flow is a non-constant solution with $\phi_T(x) = x$ for some $T > 0$ . A limit cycle is a periodic orbit that is isolated: some neighbourhood contains no other periodic orbit. The ω-limit set $\omega(x)$ is the set of points $y$ with $\phi_{t_k}(x) \to y$ for some sequence $t_k \to \infty$ — where the orbit ends up.

Definition— conjugacy, bifurcation

Maps $f : X \to X$ and $g : Y \to Y$ are topologically conjugate if there is a homeomorphism $h : X \to Y$ with $h \circ f = g \circ h$ . Then $h$ carries orbits to orbits, so the two systems are "the same up to a change of coordinates". If $h$ is only a continuous surjection, $g$ is a factor of $f$ (a semi-conjugacy). A bifurcation is a qualitative change — in the number or stability of fixed points or cycles, say — in a family $f_\mu$ as the parameter $\mu$ varies.

Definition— sensitive dependence and chaos

$f : X \to X$ has sensitive dependence on initial conditions if there is $\delta > 0$ such that for every $x$ and every $\varepsilon > 0$ there are $y$ with $d(x, y) < \varepsilon$ and $n \ge 0$ with $d(f^n(x), f^n(y)) \ge \delta$ . Following Devaney (1986), $f$ is chaotic on $X$ if it has sensitive dependence, its periodic points are dense, and it is topologically transitive: for all non-empty open $U, V$ there is $n$ with $f^n(U) \cap V \ne \varnothing$ .

The Lyapunov exponent of an orbit of a $C^1$ map of $\R$ is $\lambda(x) = \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\ln|f'(f^k(x))|$ , when the limit exists. It is the average exponential rate at which nearby orbits separate. Positive exponents are the usual numerical signature of chaos.

Worked examplesH

Examples

Example— Linear maps of the line

For $f(x) = ax$ , $f^n(x) = a^nx$ . The fixed point $0$ attracts if $|a| < 1$ and repels if $|a| > 1$ . If $a < 0$ the orbit alternates sides. For $|a| = 1$ nothing moves closer or further: $a = -1$ makes every non-zero point periodic of period $2$ .

Example— The doubling map and the circle

$D(x) = 2x \bmod 1$ on $[0, 1)$ , which we think of as the circle $\R/\Z$ . In binary, $D$ deletes the first digit: $D(0.b_1b_2b_3\ldots) = 0.b_2b_3\ldots$ . So $x$ is periodic exactly when its binary expansion repeats, and the orbit of a point whose binary digits are "random" visits every interval. Points that agree in their first $n$ digits are within $2^{-n}$ , but after $n$ steps their orbits have nothing in common.

Example— An irrational rotation: dense orbits without chaos

$R_\alpha(x) = x + \alpha \bmod 1$ with $\alpha$ irrational. Every orbit is dense in the circle (Kronecker), so $R_\alpha$ is transitive. But $R_\alpha$ is an isometry, so nearby points stay exactly as close as they were: no sensitive dependence, and no periodic points at all. Transitivity alone is not chaos.

Example— The pendulum

$\ddot\theta = -\sin\theta$ , or $\dot\theta = v$ , $\dot v = -\sin\theta$ . The energy $E = \frac12v^2 - \cos\theta$ is constant along orbits ( $\dot E = v\dot v + \sin\theta\,\dot\theta = 0$ ), so orbits lie on its level curves. The equilibrium $(0, 0)$ is surrounded by closed orbits (a centre), while $(\pi, 0)$ is a saddle. The orbits through the saddle, the separatrices, divide swinging from spinning.

ProofH

Fixed points of maps and the logistic family

Theorem— Theorem 1 (derivative test)

Let $f : \R \to \R$ be $C^1$ with $f(x^*) = x^*$ . If $|f'(x^*)| < 1$ , then $x^*$ is attracting. If $|f'(x^*)| > 1$ , then $x^*$ is repelling.

Proof

Suppose $|f'(x^*)| < 1$ and pick $K$ with $|f'(x^*)| < K < 1$ . By continuity of $f'$ there is $\delta > 0$ with $|f'| \le K$ on $I = [x^* - \delta, x^* + \delta]$ . For $x \in I$ the mean value theorem gives $|f(x) - x^*| = |f(x) - f(x^*)| = |f'(\xi)||x - x^*| \le K|x - x^*|$ . In particular $f(x) \in I$ , so by induction $f^n(x) \in I$ and $|f^n(x) - x^*| \le K^n|x - x^*| \to 0$ .

If $|f'(x^*)| > 1$ , choose $K > 1$ and $\delta$ with $|f'| \ge K$ on $I$ . As long as $f^n(x)$ stays in $I$ , $|f^{n+1}(x) - x^*| \ge K|f^n(x) - x^*|$ . Since $K^n|x - x^*| \to \infty$ for $x \ne x^*$ , every orbit except $x^*$ leaves $I$ .

∎

When $|f'(x^*)| = 1$ the test says nothing: $x \mapsto x - x^3$ attracts to $0$ , $x \mapsto x + x^3$ repels, and both have $f'(0) = 1$ . The same proof works for periodic orbits, using the multiplier $(f^n)'$ .

Theorem— Theorem 2 (the logistic map up to the first period doubling)

Let $f_r(x) = rx(1 - x)$ with $r > 0$ .

(a) The fixed points are $0$ and $x_r^* = 1 - \frac1r$ , with $f_r'(0) = r$ and $f_r'(x_r^*) = 2 - r$ . So $0$ attracts for $r < 1$ , and $x_r^*$ attracts for $1 < r < 3$ and repels for $r > 3$ .

(b) For $r > 3$ there is an orbit of minimal period $2$ , $\{p_+, p_-\}$ with $p_\pm = \frac{r + 1 \pm \sqrt{(r - 3)(r + 1)}}{2r}$ . Its multiplier is $f_r'(p_+)f_r'(p_-) = 4 + 2r - r^2$ , so it is attracting for $3 < r < 1 + \sqrt6 \approx 3.449$ .

Proof

(a) $rx(1 - x) = x$ iff $x(r - 1 - rx) = 0$ . The derivative is $f_r'(x) = r(1 - 2x)$ , which at $x^*_r$ equals $r\big(\frac2r - 1\big) = 2 - r$ . Now $|2 - r| < 1$ iff $1 < r < 3$ , and Theorem 1 applies.

(b) Period-2 points satisfy $f_r(f_r(x)) = x$ , a quartic divisible by the fixed-point quadratic $x(rx - r + 1)$ . The quotient is $r^2x^2 - r(r+1)x + (r + 1) = 0$ , with roots $p_\pm$ , real and distinct exactly when $(r - 3)(r + 1) > 0$ , that is $r > 3$ . With $s = \sqrt{(r-3)(r+1)}$ one checks directly that $f_r(p_+) = p_-$ and $f_r(p_-) = p_+$ (Lean verifies the algebra below). Since $p_+ \ne p_-$ , the minimal period is $2$ . For the multiplier, $p_+ + p_- = \frac{r + 1}{r}$ and $p_+p_- = \frac{r + 1}{r^2}$ , so

r^2(1 - 2p_+)(1 - 2p_-) = r^2\big(1 - 2(p_+ + p_-) + 4p_+p_-\big) = r^2 - 2r(r + 1) + 4(r + 1) = 4 + 2r - r^2.

This is less than $1$ iff $r^2 - 2r - 3 > 0$ , i.e. $r > 3$ , and greater than $-1$ iff $r^2 - 2r - 5 < 0$ , i.e. $r < 1 + \sqrt6$ .

∎

At $r = 3$ the multiplier of $x^*_r$ passes through $-1$ . The fixed point loses stability and hands it to a newly born 2-cycle: a period-doubling bifurcation. At $r = 1 + \sqrt6$ the 2-cycle's multiplier reaches $-1$ and a stable 4-cycle is born, and so on. The doubling parameters $r_1 = 3$ , $r_2 \approx 3.4495$ , $r_3 \approx 3.5441$ , $r_4 \approx 3.5644, \dots$ accumulate at $r_\infty \approx 3.5699$ . Mitchell Feigenbaum discovered numerically in 1975 that the gaps shrink geometrically,

\frac{r_{n} - r_{n-1}}{r_{n+1} - r_n} \to \delta = 4.6692016\ldots,

and that the same $\delta$ appears for every smooth unimodal family with a quadratic maximum, such as $x \mapsto r\sin(\pi x)$ . This universality was explained by renormalisation: Feigenbaum (1978) and, independently, Coullet and Tresser. It was proved by Oscar Lanford with a computer-assisted argument in 1982, and Mikhail Lyubich (1999) gave a conceptual proof of the renormalisation conjectures. The experiment below estimates $\delta$ . That estimate is evidence, not a proof.

Beyond $r_\infty$ there are windows of periodic behaviour, the largest beginning at $r = 1 + \sqrt8 \approx 3.8284$ with an attracting 3-cycle. Sharkovskii's theorem (1964) orders the positive integers as

3 \rhd 5 \rhd 7 \rhd \cdots \rhd 2\cdot3 \rhd 2\cdot5 \rhd \cdots \rhd 2^2\cdot3 \rhd \cdots \rhd 2^3 \rhd 2^2 \rhd 2 \rhd 1,

and states that a continuous map of an interval with a periodic point of period $m$ has periodic points of every period $n$ with $m \rhd n$ . In particular period three implies all periods (Li and Yorke, 1975, who also coined the word "chaos" in this sense).

ProofH

Planar flows: linearisation and Lyapunov functions

For a linear system $\dot x = Ax$ with $A$ a real $2\times2$ matrix, the solution is $x(t) = e^{tA}x(0)$ . Its behaviour depends on the eigenvalues, which are determined by $\tau = \operatorname{tr}A$ and $\Delta = \det A$ : they are $\lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}$ .

Theorem— Theorem 3 (classification of planar linear systems)

Let $\Delta \ne 0$ , so that $0$ is the only equilibrium.

$\Delta < 0$ : a saddle (real eigenvalues of opposite signs; unstable).
$\Delta > 0$ , $\tau^2 \ge 4\Delta$ : a node (real eigenvalues of the same sign), stable if $\tau < 0$ and unstable if $\tau > 0$ .
$\Delta > 0$ , $\tau^2 < 4\Delta$ , $\tau \ne 0$ : a spiral (complex eigenvalues $\alpha \pm i\beta$ , $\alpha = \tau/2$ ), stable if $\tau < 0$ and unstable if $\tau > 0$ .
$\Delta > 0$ , $\tau = 0$ : a centre (eigenvalues $\pm i\beta$ ): every other orbit is a closed ellipse.

In particular $0$ is asymptotically stable if and only if $\tau < 0 < \Delta$ .

Proof

Since $\lambda_1\lambda_2 = \Delta$ and $\lambda_1 + \lambda_2 = \tau$ , the sign cases follow from the formula for the eigenvalues. For real distinct eigenvalues, in an eigenbasis the system decouples as $\dot y_i = \lambda_iy_i$ , so $y_i(t) = e^{\lambda_it}y_i(0)$ . Both decay if both $\lambda_i < 0$ , both grow if both are positive, and one grows while the other decays if they have opposite signs (the saddle). For a repeated eigenvalue $\lambda$ that is not diagonalisable, the Jordan form gives $y_1 = e^{\lambda t}(y_1(0) + ty_2(0))$ , $y_2 = e^{\lambda t}y_2(0)$ , which still decays exactly when $\lambda < 0$ . For complex eigenvalues $\alpha \pm i\beta$ , there is a real basis in which $A = \begin{pmatrix}\alpha & -\beta\\ \beta & \alpha\end{pmatrix}$ . In polar coordinates this is $\dot r = \alpha r$ , $\dot\theta = \beta$ , so $r = e^{\alpha t}r(0)$ : inward spirals for $\alpha < 0$ , outward for $\alpha > 0$ , and circles for $\alpha = 0$ (ellipses in the original coordinates).

∎

For a nonlinear system $\dot x = F(x)$ with $F(x^*) = 0$ , the linearisation $\dot y = DF(x^*)y$ describes the motion near $x^*$ — when $x^*$ is hyperbolic.

Theorem— Theorem 4 (Hartman–Grobman; stated, not proved here)

If $x^*$ is a hyperbolic equilibrium of a $C^1$ flow, there is a homeomorphism between neighbourhoods of $x^*$ and of $0$ that carries orbits of the flow to orbits of its linearisation, preserving the direction of time. In particular, if all eigenvalues of $DF(x^*)$ have negative real part, $x^*$ is asymptotically stable.

(Proofs: Hartman 1960 and Grobman 1959; see Perko, Differential Equations and Dynamical Systems, §2.8.) Hyperbolicity cannot be dropped. The systems

\dot x = -y \mp x(x^2 + y^2), \qquad \dot y = x \mp y(x^2 + y^2)

both have the centre $\dot x = -y$ , $\dot y = x$ as linearisation. In polar coordinates they are $\dot r = \mp r^3$ , $\dot\theta = 1$ . With the upper sign the origin is an asymptotically stable spiral, and with the lower sign an unstable one. A linear centre decides nothing.

Theorem— Theorem 5 (Lyapunov's direct method)

Let $x^*$ be an equilibrium of $\dot x = F(x)$ and $V$ a $C^1$ function on a neighbourhood $U$ of $x^*$ with $V(x^*) = 0$ and $V > 0$ on $U \setminus \{x^*\}$ . If $\dot V = \nabla V \cdot F \le 0$ on $U$ , then $x^*$ is Lyapunov stable. If $\dot V < 0$ on $U \setminus \{x^*\}$ , then $x^*$ is asymptotically stable.

Proof

Take $\varepsilon > 0$ with the closed ball $\overline{B}(x^*, \varepsilon) \subseteq U$ , and let $m = \min V$ on the sphere $|x - x^*| = \varepsilon$ , so $m > 0$ . By continuity there is $\delta \in (0, \varepsilon)$ with $V < m$ on $B(x^*, \delta)$ . Since $V$ does not increase along orbits, an orbit starting in $B(x^*, \delta)$ keeps $V < m$ and so never reaches the sphere: it stays in $B(x^*, \varepsilon)$ . That is stability.

Now suppose $\dot V < 0$ away from $x^*$ . Along such an orbit, $V(\phi_t(x))$ decreases to a limit $c \ge 0$ . If $c > 0$ , the orbit stays in the compact set $\{c \le V,\ |x - x^*| \le \varepsilon\}$ , on which $\dot V \le -\eta < 0$ for some $\eta$ . Then $V \to -\infty$ , which is absurd. So $V(\phi_t(x)) \to 0$ , which forces $\phi_t(x) \to x^*$ (if not, the orbit would return infinitely often to a compact set avoiding $x^*$ , on which $V$ is bounded below by a positive constant).

∎

For the spiral sink $\dot x = -x + y$ , $\dot y = -x - y$ , the function $V = x^2 + y^2$ gives $\dot V = 2x(-x + y) + 2y(-x - y) = -2V$ , so $V$ decays like $e^{-2t}$ (formalised below). For the systems above with the upper sign, $V = x^2 + y^2$ gives $\dot V = -2(x^2 + y^2)^2 < 0$ , which proves the asymptotic stability that linearisation could not see.

ProofH

Limit cycles and the Poincaré–Bendixson theorem

The simplest limit cycle is visible in polar coordinates: $\dot r = r(1 - r^2)$ , $\dot\theta = 1$ . For $0 < r < 1$ , $r$ increases, and for $r > 1$ it decreases, so every orbit except the origin spirals onto the unit circle. Usually no such coordinates exist, and the existence of a cycle needs a theorem.

Theorem— Theorem 6 (Poincaré–Bendixson; stated with a proof sketch)

Let $\dot x = F(x)$ be a $C^1$ flow on an open set of $\R^2$ , and suppose the forward orbit of $x$ stays in a compact set $K$ that contains no equilibrium. Then $\omega(x)$ is a periodic orbit. Either the orbit of $x$ is periodic, or it spirals towards a periodic orbit.

Proof sketch. $\omega(x)$ is non-empty, compact, connected and invariant. Take $y \in \omega(x)$ and $z \in \omega(y) \subseteq \omega(x)$ . Since $F(z) \ne 0$ , there is a short segment $\Sigma$ through $z$ transverse to the flow. The key lemma: the successive crossings of $\Sigma$ by any orbit are monotone along $\Sigma$ . A piece of orbit between two crossings, closed up by a piece of $\Sigma$ , is a simple closed curve. By the Jordan curve theorem it separates the plane, and the flow crosses $\Sigma$ in one direction only, so the orbit cannot come back on the other side. Hence an ω-limit set meets $\Sigma$ in at most one point. This forces the orbit of $y$ to return to the same point of $\Sigma$ , so it is periodic, and then $\omega(x)$ is that periodic orbit. Full proofs: Hirsch, Smale and Devaney (2013), chapter 10, or Perko (2001), §3.7. $\square$

Example— A trapping annulus

Consider $\dot r = r(1 - r^2) + \mu r\cos\theta$ , $\dot\theta = 1$ , with $0 < \mu < 1$ (Strogatz's example). Since $\dot\theta = 1$ there are no equilibria except possibly the origin. On the circle $r = r_{\min} < \sqrt{1 - \mu}$ we have $\dot r \ge r(1 - r^2 - \mu) > 0$ , and on $r = r_{\max} > \sqrt{1 + \mu}$ we have $\dot r \le r(1 - r^2 + \mu) < 0$ . So the annulus $r_{\min} \le r \le r_{\max}$ is trapping: orbits enter it and never leave. It is compact and contains no equilibrium, so by Poincaré–Bendixson it contains a periodic orbit — although we cannot write it down.

Theorem— Corollary 7 (no chaos in the plane)

A bounded orbit of a planar $C^1$ flow with finitely many equilibria converges to an equilibrium, to a periodic orbit, or to a cycle of equilibria joined by orbits. In particular planar flows have no strange attractors and no sensitive dependence of the kind seen in the doubling map.

(The generalised Poincaré–Bendixson theorem gives exactly this trichotomy; see Perko, §3.7.) Chaos for flows needs at least three dimensions. Lorenz's system $\dot x = \sigma(y - x)$ , $\dot y = x(\rho - z) - y$ , $\dot z = xy - \beta z$ with $\sigma = 10$ , $\rho = 28$ , $\beta = 8/3$ has a famous butterfly-shaped attractor. Only in 2002 did Warwick Tucker prove, with rigorous computer-assisted interval arithmetic, that it really is a strange attractor (a problem on Smale's 1998 list). For maps, one dimension is enough: the logistic map shows it.

Bifurcations of flows. On the line, the normal forms are the saddle-node $\dot x = \mu + x^2$ (two equilibria collide and vanish as $\mu$ increases through $0$ ), the transcritical $\dot x = \mu x - x^2$ (two equilibria exchange stability) and the pitchfork $\dot x = \mu x - x^3$ (one equilibrium splits into three). In the plane, in a Hopf bifurcation a pair of complex eigenvalues crosses the imaginary axis and a small limit cycle is born — for instance $\dot r = \mu r - r^3$ , $\dot\theta = 1$ , which has the cycle $r = \sqrt\mu$ for $\mu > 0$ . A precise statement, with its non-degeneracy conditions, is in Strogatz (2015), chapter 8.

ProofH

Chaos, proved

Theorem— Theorem 8 (the doubling map is chaotic)

$D(x) = 2x \bmod 1$ on the circle $\R/\Z$ (with distance $d(x, y) =$ the shorter arc) has sensitive dependence with $\delta = \frac14$ , dense periodic points, and is topologically transitive.

Proof

Expansion. If $d(x, y) < \frac14$ , then $d(D(x), D(y)) = 2d(x, y)$ .

Sensitive dependence. Let $x$ and $\varepsilon > 0$ be given, and let $y \ne x$ with $d(x, y) < \min(\varepsilon, \frac14)$ . While the distance is below $\frac14$ , it doubles at each step. So there is a first $n$ with $d(D^n(x), D^n(y)) \ge \frac14$ .

Dense periodic points. $D^n(x) = 2^nx \bmod 1$ , so $D^n(x) = x$ iff $(2^n - 1)x \in \Z$ . The periodic points of period $n$ are the $2^n - 1$ points $\frac{k}{2^n - 1}$ , spaced $\frac{1}{2^n - 1}$ apart. Every interval contains one for large $n$ .

Transitivity. An open set $U$ contains an arc $J = \big[\frac{k}{2^n}, \frac{k+1}{2^n}\big]$ for large $n$ , and $D^n$ maps $J$ onto the whole circle. So $D^n(U)$ meets every $V$ .

∎

Theorem— Theorem 9 (the logistic map at r = 4 is a factor of the doubling map)

Let $h(\theta) = \sin^2(\pi\theta)$ , a continuous surjection from the circle $\R/\Z$ onto $[0, 1]$ . Then $f_4 \circ h = h \circ D$ . Consequently $f_4$ has dense periodic points and is topologically transitive on $[0, 1]$ , hence chaotic in Devaney's sense, and it has periodic points of every period.

Proof

$f_4(h(\theta)) = 4\sin^2(\pi\theta)\cos^2(\pi\theta) = \sin^2(2\pi\theta) = h(D(\theta))$ . (The identity is well defined on the circle because $\sin^2$ has period $\pi$ .) If $D^n(\theta) = \theta$ , then $f_4^n(h(\theta)) = h(D^n(\theta)) = h(\theta)$ . So $h$ maps the dense set of periodic points of $D$ to a set of periodic points of $f_4$ , which is dense because $h$ is a continuous surjection. For transitivity, given open $U, V \subseteq [0, 1]$ , the sets $h^{-1}(U)$ and $h^{-1}(V)$ are open and non-empty, so $D^n(h^{-1}U)$ meets $h^{-1}V$ for some $n$ , and applying $h$ shows that $f_4^n(U)$ meets $V$ . Sensitive dependence then follows from transitivity and dense periodic points on an infinite metric space (Banks, Brooks, Cairns, Davis and Stacey, 1992).

∎

So the logistic map at $r = 4$ is chaotic: not "according to the computer", but by a proof. The conjugacy also gives exact formulas. The orbit of $x_0 = \sin^2(\pi\theta_0)$ is $x_n = \sin^2(2^n\pi\theta_0)$ , and the Lyapunov exponent of almost every orbit is exactly $\ln 2$ . For most $r$ in $(r_\infty, 4)$ , by contrast, what happens is subtle. The parameters with an attracting cycle are dense (Graczyk–Świątek and Lyubich, 1997), yet the parameters with an absolutely continuous invariant measure — "stochastic" behaviour — have positive measure (Jakobson, 1981).

Formal mathematicsF

Dynamics in Lean and Mathlib

Mathlib's Dynamics library provides the vocabulary: iterates f^[n], Function.IsFixedPt, Function.IsPeriodicPt and Function.minimalPeriod, flows, ω-limit sets, and the Banach fixed point theorem. The lesson's concrete theorems about the logistic map are proved here by algebra and calculus.

logisticLean 4Formally verified

/-- The **logistic map** `f_r(x) = r x (1 - x)`, a model of a population with limited resources
(`x` is the population as a fraction of its maximum, `r` the growth rate). -/
def logistic (r x : ℝ) : ℝ := r * x * (1 - x)