Discrete Maps and the Logistic Map

In 1976, Robert May published a plea in Nature with an unusual thesis: the most important equation a student could meet was one line long. "Simple mathematical models with very complicated dynamics" argued that the humble recursion $x_{n+1} = r\,x_n(1 - x_n)$ — a population model naive enough to sketch on a napkin — contained steady states, oscillations of every period, and fully developed chaos, and that anyone who understood this one parabola would be inoculated against the deepest scientific superstition: that complicated behaviour requires complicated causes. May ended by proposing that the map be taught early and universally. This chapter ends by obeying. The logistic map is the chaos block's entire curriculum in miniature, run in the cheapest possible arena. Discrete time replaces flows — no integrator, no phase-space dimension count, just iteration — yet nothing essential is lost, because a one-dimensional map is exactly what a strongly dissipative flow looks like through a Poincaré section, as the previous article established. Stability becomes a single derivative measured against 1 rather than 0; bifurcations happen at multiplier $+1$ and $-1$ ; cobweb diagrams replace phase portraits; and the entire route to chaos — cascade, Feigenbaum universality, positive Lyapunov exponent — plays out along one parameter axis from $r = 0$ to $4$ . What this closing article adds to the chapter's earlier visits (the cascade in 0231, the constants in 0232, $\lambda = \ln 2$ in 0235) is the map's complete biography: the extinction and steady-state regimes with their transcritical handoff, the cobweb geometry of iteration, the fully chaotic endpoint $r = 4$ where the dynamics is exactly solvable — a closed-form chaotic solution, with an exact invariant density and ergodic averages — and the strange interior of the chaotic regime: bands, intermittency, and the period-3 window whose birth at $r = 1 + \sqrt8$ carries Sharkovskii's astonishing theorem behind it.

Key takeaways

A discrete map $x_{n+1} = f(x_n)$ advances the state by iteration; the logistic map $x_{n+1} = r\,x_n(1 - x_n)$ on $[0,1]$ , $0 \le r \le 4$ , is the canonical one-hump specimen.
Fixed points obey $x^* = f(x^*)$ ; stability is set by the multiplier $\mu = f'(x^*)$ : stable iff $|\mu| < 1$ , with monotone approach for $\mu > 0$ and alternating approach for $\mu < 0$ .
Biography: extinction ( $x^* = 0$ stable) for $r < 1$ ; transcritical exchange at $r = 1$ ; stable steady state $x^* = 1 - 1/r$ for $1 < r < 3$ ; period-doubling cascade from $r = 3$ accumulating at $r_\infty = 3.5699$ ; chaos beyond, riddled with periodic windows.
At $r = 4$ the chaos is exactly solvable: $x_n = \sin^2(2^n\pi\theta_0)$ — a closed-form solution with $\lambda = \ln 2$ and invariant density $\rho(x) = 1/[\pi\sqrt{x(1-x)}]$ .
The period-3 window opens by a tangent (saddle-node) bifurcation at $r = 1 + \sqrt8 = 3.8284$ , preceded by intermittency; "period three implies chaos" (Li–Yorke) and Sharkovskii's ordering organize the window zoo.
One-dimensional maps escape the planar no-chaos theorems because iteration has no continuity-of-trajectory constraint — the fold does what flows need three dimensions to do.
Key idea: stability of a discrete map's fixed point is governed by the magnitude of the multiplier $|f'(x^*)|$ — stable if below one, unstable if above — and the marginal value $f'(x^*) = -1$ triggers period-doubling, launching a universal cascade to chaos governed by the Feigenbaum constant.

The core insight

Stability of a discrete map's fixed point is governed by the magnitude of the multiplier $|f'(x^*)|$ : stable if $<1$ , unstable if $>1$ . The marginal value $f'(x^*)=-1$ triggers period-doubling, launching a universal cascade to chaos governed by the Feigenbaum constant.

Where this lives today

The logistic map is now infrastructure in three separate ways. As pedagogy, May's plea won completely: the map is the standard first encounter with chaos from high-school enrichment to graduate courses, and the bifurcation diagram of Figure 2 is arguably the most reproduced image in nonlinear science. As a research testbed, it remains the sharpest available whetstone: theorems about interval maps (kneading theory, Sharkovskii-type results, the Milnor–Thurston machinery), rigorous results on when unimodal maps have absolutely continuous invariant measures (Jakobson's theorem: chaotic parameters form a positive-measure set — "chaos is not a knife-edge accident"), and modern computer-assisted proofs all cut their teeth here. As a modelling grammar, its descendants run through population biology (Ricker and Beverton–Holt maps in fisheries management, where the boom-bust of Worked Example 1 is an operational concern), epidemiology's discrete-generation models, and the analysis of anything sampled once per cycle — which, per the Poincaré-section article, is ultimately everything periodic-driven. And with it, the M10 chaos block closes its dissipative arc. The track's remaining articles turn to the Hamiltonian side — KAM theory and the standard map, where the fold is forbidden, tori resist, and chaos must negotiate with conservation laws. The parabola's lesson travels there intact: simple rules, complicated dynamics, and a small set of universal numbers governing the border between them.

The physical picture

Iteration has a geometry you can do by hand. Draw the parabola $y = f(x)$ and the diagonal $y = x$ on the same axes. From $x_0$ , move vertically to the curve — that is $x_1 = f(x_0)$ — then horizontally to the diagonal to reset the input, then vertically again. The staircase-spiral this traces is the cobweb, and it makes the multiplier criterion visible: where the parabola crosses the diagonal shallowly ( $|f'| < 1$ ) the cobweb spirals in and the crossing is a stable fixed point; where the crossing is steep ( $|f'| > 1$ ) the cobweb unwinds outward. A negative slope makes the cobweb a true spiral — alternating sides, the overshoot dynamics whose marginal case $f' = -1$ launched the doubling cascade two articles ago. In the chaotic regime the cobweb never settles: it fills a region of the square, bouncing between the arms of the parabola forever.

Figure

Cobweb plots — the geometry of iteration, converging at small r and wandering forever in the chaotic regime

The parabola's fold is the engine. The map sends both $x$ and $1 - x$ to the same image: the interval is stretched (slope up to $r$ in magnitude) and folded back onto itself each step — the baker's kneading met in the sensitive-dependence article, now in its minimal form. This is also why a one-dimensional map can be chaotic while a one-dimensional flow never can: a flow's state slides continuously and cannot pass through itself, but iteration hops, and the fold costs nothing. The Poincaré-section article supplies the physical pedigree — a strongly contracting 3D flow leaves a section map that is effectively one-dimensional and one-humped — so nothing here is a toy: this is the driven pendulum's stroboscope with the inessentials boiled away. Run the parameter and the whole chapter replays. For $r < 1$ the parabola lies below the diagonal: every cobweb slides to extinction at $x^* = 0$ . At $r = 1$ the two fixed points collide and exchange stability — a transcritical bifurcation, the ecology-flavoured member of the bifurcation zoo from the start of this track. For $1 < r < 3$ the population equilibrates at $x^* = 1 - 1/r$ , approaching monotonically for $r < 2$ and with alternating overshoot for $2 < r < 3$ . At $r = 3$ the multiplier hits $-1$ : the flip, the cascade, $r_\infty = 3.569946$ , Feigenbaum — all of it as covered, and visible in one computed figure:

Figure

The complete biography of the logistic map — steady state, cascade, chaos, and the period-3 window at r equal to one plus root eight

Beyond $r_\infty$ the diagram's dark bands are the new territory this article owns. The chaos is structured: bands that merge pairwise as $r$ grows (an inverse cascade, paced by the same $\delta$ ), and bright vertical windows where stable periodic orbits are reborn — most prominently the period-3 window at $r \approx 3.83$ , whose opening mechanism (a tangent bifurcation, with intermittency just below it) is Derivation C. And at the right-hand edge sits the miracle: $r = 4$ , where the fully chaotic dynamics admits a closed-form solution. Chaos with a formula — the cleanest possible demonstration that deterministic, solvable, and predictable are three different words.

The map on its own terms

The general object and its specimen:

x_{n+1} = f(x_n), \qquad x_{n+1} = r\,x_n(1 - x_n) \qquad (1)

where: $x_n \in [0,1]$ — the state (a normalized population: growth $\propto x$ , resources $\propto 1 - x$ ); $r \in [0,4]$ — the single control parameter. The ceiling $r \le 4$ is forced by self-consistency: the parabola's maximum is $f(1/2) = r/4$ , and for $r > 4$ iterates can exit $[0,1]$ and escape to $-\infty$ — the map must send the interval into itself for the dynamics to live. Fixed points solve $x^* = f(x^*)$ :

x^* = 0 \qquad \text{and} \qquad x^* = 1 - \frac{1}{r} \quad (\text{in } [0,1] \text{ for } r \ge 1) \qquad (2)

where: the two equilibria are extinction and steady coexistence. Their fates are governed by the map analogue of the eigenvalue — the multiplier. Linearizing $x_n = x^* + \delta x_n$ gives $\delta x_{n+1} = \mu\,\delta x_n$ with

\mu = f'(x^*), \qquad |\delta x_n| = |\mu|^n\,|\delta x_0|, \qquad \text{stable} \iff |\mu| < 1 \qquad (3)

where: the stability boundary is the unit circle for maps, not the imaginary axis — perturbations are multiplied per step, not exponentiated per unit time. The sign of $\mu$ is dynamical information the continuous world lacks: $0 < \mu < 1$ gives monotone settling, $-1 < \mu < 0$ alternating settling — the cobweb spiral. For the logistic map, $f'(x) = r(1 - 2x)$ , so

f'(0) = r, \qquad f'\big(1 - \tfrac1r\big) = 2 - r \qquad (4)

where: the extinction point is stable for $r < 1$ and hands over at $r = 1$ (transcritical — the two fixed points cross and trade stability, Derivation A); the coexistence point is stable for $|2 - r| < 1$ , i.e. $1 < r < 3$ , with the character change at $r = 2$ (monotone to alternating) and the flip at $r = 3$ . Periodic orbits are fixed points of iterates: a $k$ -cycle $\{p_1, \ldots, p_k\}$ satisfies $f^{(k)}(p_i) = p_i$ , and its multiplier is the chain-rule product around the loop,

\mu_k = \prod_{i=1}^{k} f'(p_i) \qquad (5)

where: the product structure — one factor per visit — is what made the cascade computable in article 0231 and the Lyapunov exponent an orbit average in 0236. The cascade thresholds $r_1 = 3$ , $r_2 = 1 + \sqrt6$ , $r_\infty = 3.569946$ and the universal $\delta = 4.669$ are that machinery's output, imported here rather than re-derived. Two quantities summarize the chaotic regime. The Lyapunov exponent of the map,

\lambda = \lim_{N\to\infty}\frac1N \sum_{n=0}^{N-1}\ln\big|f'(x_n)\big| \qquad (6)

where: negative through every periodic regime, zero at the thresholds, positive in the chaotic bands (the computed $\lambda(r)$ curve lives in the Lyapunov article). And the invariant density $\rho(x)$ — the long-run histogram of a chaotic orbit — which at $r = 4$ is known exactly:

\rho(x) = \frac{1}{\pi\sqrt{x(1 - x)}} \qquad (7)

where: the density diverges (integrably) at the endpoints — the orbit lingers near $0$ and $1$ because the map's images pile up under the fold — and time averages of any observable equal $\int g(x)\rho(x)\,\mathrm{d}x$ : ergodicity, checkable to three digits on a laptop (Figure 3 below). The exactness comes from the conjugacy underlying Derivation B, which also delivers the closed-form solution and $\lambda = \ln 2$ .

Figure

The invariant density of the fully chaotic map — time averages converging to the exact arcsine law

Which tool answers which question

you want	use	notes
iterate by eye	cobweb construction	shallow crossing = stable; steep = unstable
fixed-point fate	multiplier (3)–(4) vs the unit circle	sign gives monotone vs alternating
cycle stability	chain-rule product (5)	one slope factor per cycle point
where the cascade sits	$r_1 = 3$ , $r_\infty = 3.5699$ , $\delta = 4.669$	owned by articles 0231–0232
is this $r$ chaotic?	$\lambda(r)$ , eq. (6)	windows dip negative inside chaos
long-run statistics at $r = 4$	density (7) + closed form (Derivation B)	exact ergodic averages
why a window just opened	tangent bifurcation + intermittency	Derivation C; period-3 at $1 + \sqrt8$

Derivation AThe transcritical handoff and the life of the steady state

We derive the map's first two acts: the exchange of stability between extinction and coexistence at $r = 1$ , and the full stability window of the nonzero fixed point with its qualitative substructure. The strategy is elementary multiplier bookkeeping — but done completely, because this is the template every later bifurcation reuses. The step to watch is the transcritical mechanism: unlike a saddle-node, no fixed point is born or dies at $r = 1$ ; two permanent residents collide and swap jobs, and the negative-population branch quietly bookkeeps the exchange.

Step 1. Locate the fixed points for all $r$ : $x^* = 0$ always, and $x^* = 1 - 1/r$ , which lies in the physical interval only for $r \ge 1$ (for $r < 1$ it sits at negative $x$ — mathematically present, ecologically meaningless).

Step 2. Multipliers from $f'(x) = r(1 - 2x)$ : at extinction $\mu_0 = r$ ; at coexistence $\mu_* = 2 - r$ . For $r < 1$ : $\mu_0 = r < 1$ , extinction stable — any small population dies out ( $f(x) \approx rx$ with $r < 1$ is per-step decay). The coexistence point, where it lurks at $x^* < 0$ , has $|\mu_*| = |2 - r| > 1$ : unstable.

Step 3. Collide at $r = 1$ : the two fixed points coincide ( $1 - 1/r \to 0$ ) with common multiplier $\mu = 1$ — the $+1$ boundary of the unit circle, the map analogue of a zero eigenvalue. For $r > 1$ they separate again with roles exchanged: $\mu_0 = r > 1$ (extinction now repels — a small population grows) and $\mu_* = 2 - r \in (-1, 1)$ (coexistence attracts). Why this is transcritical and not saddle-node: both fixed points exist on both sides of the bifurcation; only stability is traded. The signature in formulas is the multiplier of each passing through $+1$ at the collision; the signature in ecology is a sharp threshold — below unit reproductive rate, extinction is the attractor no matter the starting population.

Step 4. Read the coexistence window $1 < r < 3$ with its interior structure. For $1 < r < 2$ : $\mu_* \in (0,1)$ — monotone relaxation to equilibrium (cobweb staircase). At $r = 2$ : $\mu_* = 0$ — the superstable case, equilibrium reached quadratically fast (the fixed point sits exactly at the parabola's maximum $x^* = 1/2$ ). For $2 < r < 3$ : $\mu_* \in (-1, 0)$ — alternating convergence, populations overshooting the equilibrium in damped oscillation each generation.

Step 5. End the act at $r = 3$ : $\mu_* = -1$ , the flip threshold — perturbations alternate without decay, and the 2-cycle takes the stage. From here the story is the cascade article's; this derivation is the platform it stood on. The multipliers follow by direct differentiation; superstability at $r = 2$ is confirmed by noting that $x^* = 1/2$ there and $f'(1/2) = 0$ .

Go deeper · Derivation B · 2 minChaos in closed form — the exact solution at r equals four

Derivation BChaos in closed form — the exact solution at r equals four

The goal is the most surprising formula in this chapter: an explicit closed-form solution of a fully chaotic system, $x_n = \sin^2(2^n\pi\theta_0)$ , together with its three payoffs — the digit-shift picture of chaos, the exact Lyapunov exponent, and the exact invariant density (7). The strategy is a change of variables that straightens the parabola's fold into a doubling of angle. The step to watch is how a solvable system can still be unpredictable: the formula is exact, and useless, and the reason why is the whole philosophy of the chapter in one line.

Step 1. Substitute $x_n = \sin^2(\pi\theta_n)$ with $\theta_n \in [0,1]$ . Then

x_{n+1} = 4x_n(1 - x_n) = 4\sin^2(\pi\theta_n)\cos^2(\pi\theta_n) = \sin^2(2\pi\theta_n)

so the map in the angle variable is the doubling map $\theta_{n+1} = 2\theta_n \pmod 1$ . Why the fold became a doubling: the substitution wraps the interval around a half-circle; the parabola's two-to-one fold is exactly the two-to-one wrapping of angle doubling. The conjugacy is smooth except at the endpoints — enough to transport all the statistics.

Step 2. Solve the doubling map by inspection: $\theta_n = 2^n\theta_0 \pmod 1$ , hence the closed form

x_n = \sin^2\big(2^n \pi\,\theta_0\big), \qquad \theta_0 = \tfrac{1}{\pi}\arcsin\sqrt{x_0}

An explicit, non-recursive formula for every iterate of a chaotic system — checkable digit for digit against direct iteration (Worked Example 2 does).

Step 3. Read chaos off the formula. Write $\theta_0$ in binary: doubling mod 1 shifts the binary expansion left one digit per step, discarding the leading bit. After $n$ steps, the $n$ -th binary digit of the initial condition has become the leading behaviour: sensitive dependence is literally digit excavation, at one bit per iteration. Two initial angles differing in the 30th bit — a difference of $10^{-9}$ — produce unrelated orbits after 30 steps. Why solvability does not rescue prediction: to use the formula for $n$ steps you must know $\theta_0$ to $n$ bits — the formula converts time-prediction into precision-demand at exactly the exponential exchange rate. Closed form and positive Lyapunov exponent coexist without tension; "unsolvable" was never what chaos meant.

Step 4. Extract the exponent exactly: the doubling map stretches uniformly by 2, so $\lambda = \ln 2$ per iteration; conjugacy invariance transports the value to the logistic map at $r = 4$ — the number that anchored the sensitive-dependence article, now with zero numerics.

Step 5. Extract the density. The doubling map preserves the uniform density in $\theta$ (each step maps the uniform measure to itself two-to-one). Push the uniform $\theta$ -measure through $x = \sin^2(\pi\theta)$ : $\rho(x) = |\mathrm{d}\theta/\mathrm{d}x| \times 1 = 1/[\pi\sqrt{x(1-x)}]$ — equation (7), the arcsine law, with its integrable spikes at the endpoints where the sine flattens. Ergodicity (of the shift map, a standard fact) transports too: time averages equal $\rho$ -weighted space averages, which is exactly what Figure 3's histogram verifies. The identity $4\sin^2 u\cos^2 u = \sin^2 2u$ is elementary; $x_5$ from the closed form matches direct iteration to 13 digits, as Worked Example 2 confirms.

Go deeper · Derivation C · 3 minThe period-3 window — tangent bifurcation, intermittency, and Sharkovskii's shadow

Derivation CThe period-3 window — tangent bifurcation, intermittency, and Sharkovskii's shadow

We derive the most conspicuous structure inside the chaotic regime: the period-3 window, born at exactly $r = 1 + \sqrt8$ . The strategy is to study the third iterate $f^{(3)}$ : a stable 3-cycle is a trio of its fixed points, and the window opens when the graph of $f^{(3)}$ becomes tangent to the diagonal — a saddle-node (tangent) bifurcation, the $+1$ exit this time, not the flip. The step to watch is what happens just below the tangency: the ghost of the not-yet-born cycle produces intermittency, a third route to chaos in this same one figure.

Step 1. Set up the third-iterate equation $f^{(3)}(x) = x$ at $r = 4 \ldots$ no — at general $r$ : a degree-8 polynomial equation whose roots include the two fixed points; the remaining six roots form two 3-cycles (each cycle contributes three roots). For small $r$ those six roots are complex: no real 3-cycle exists.

Step 2. Find the birth. As $r$ increases, the local minima and maxima of $f^{(3)}$ deepen until, at a critical $r_3$ , three of its humps touch the diagonal simultaneously (the map's symmetry gangs the tangencies together). Tangency means double roots: the six complex roots become six real ones in three coincident pairs. Carrying out the algebra (a classical computation, manageable via the trigonometric parametrization of Derivation B), the tangency condition yields exactly

r_3 = 1 + \sqrt{8} = 3.828427\ldots

Step 3. Split the pairs for $r > r_3$ : each double root separates into a stable/unstable pair — one stable 3-cycle (multiplier entering the unit circle through $+1$ ) and one unstable 3-cycle (its saddle partner, the window's basin fence). The window then runs the standard program: the stable 3-cycle period-doubles ( $3 \to 6 \to 12 \to \cdots$ ) with the same Feigenbaum $\delta$ , reaches its own accumulation, and the window closes in an interior crisis when the chaotic bands collide with the unstable 3-cycle. Why the $+1$ exit was the only way in: inside developed chaos there is no stable orbit to flip; new stability must be created from nothing, and the saddle-node is the creation event of the bifurcation dictionary — stable and unstable born together, exactly as on the section-map version in the previous article.

Step 4. Look just below $r_3$ : intermittency. For $r \lesssim r_3$ the humps of $f^{(3)}$ come close to the diagonal without touching, leaving narrow channels between curve and diagonal. An orbit entering a channel must cobweb through it in many tiny steps — during which $x_n$ cycles almost-periodically with period 3 — before escaping into a chaotic burst that eventually re-injects it into a channel. The passage time through a channel of gap $\varepsilon \propto (r_3 - r)$ scales as $T \sim \varepsilon^{-1/2}$ (the standard saddle-node ghost scaling met in the 1D-bifurcations article): as $r \to r_3^-$ , ever-longer laminar period-3 stretches interrupted by bursts — Pomeau–Manneville Type-I intermittency, the third canonical route to chaos after doubling and quasiperiodicity.

Step 5. Frame the window in the big theorems. Li and Yorke's "period three implies chaos" (1975): once a period-3 orbit exists, orbits of every period exist, along with uncountably many aperiodic orbits — inside the window all that coexists invisibly with the stable 3-cycle (everything else is unstable). Sharkovskii's 1964 ordering had said more, a decade earlier: there is a total order on the positive integers ( $3 \succ 5 \succ 7 \succ \cdots \succ 2\cdot3 \succ \cdots \succ 8 \succ 4 \succ 2 \succ 1$ ) such that a continuous interval map with a period- $m$ orbit has orbits of every period following $m$ . Period 3 sits at the top — its existence implies all periods — and the cascade's powers of two close the bottom, which is why they are the first periods to appear as $r$ grows. The window zoo of Figure 2 is Sharkovskii's ordering rendered as a bifurcation diagram. $r_3 = 1 + \sqrt8$ is the classical exact value; the tangent-channel passage time $T \sim (r_3 - r)^{-1/2}$ is the standard saddle-node ghost scaling.

Limiting cases

Case	Result	Meaning
$0 \le r < 1$	$x^* = 0$ stable ( $\mu = r$ )	extinction: sub-replacement growth
$r = 1$	transcritical, $\mu = +1$	fixed points collide and swap stability
$1 < r < 2$	$\mu_* \in (0,1)$	monotone settling to $1 - 1/r$
$r = 2$	$\mu_* = 0$ , superstable	fixed point at the maximum; quadratic convergence
$2 < r < 3$	$\mu_* \in (-1,0)$	damped alternating overshoot
$3 \le r \le r_\infty$	doubling cascade	articles 0231–0232's territory
$r_\infty < r < 4$	chaotic bands + windows	band-merging inverse cascade; period-3 window at $1+\sqrt8$
$r = 4$	fully chaotic, exactly solvable	$\lambda = \ln 2$ ; arcsine density; digit shift
$r > 4$	escape; invariant set is a Cantor repeller	orbits leak out; chaotic scattering, no attractor

Common trapMap habits and flow habits are different habits — and solvable does not mean predictable

Three transplant rejections to avoid. First, stability thresholds: for maps the boundary is the unit circle ( $|\mu| = 1$ ), not the imaginary axis; a multiplier of $-0.9$ is stable (alternating), and mistranslating flow intuition ("negative means stable") inverts verdicts. Second, dimension folklore: "chaos needs three dimensions" is a statement about continuous flows (Poincaré–Bendixson); a non-invertible 1D map folds freely and is chaotic with one variable — while an invertible 1D map never is, so keep the invertibility ledger straight when moving between maps and their suspension flows. Third, the meaning of solvable: the $r = 4$ closed form $x_n = \sin^2(2^n\pi\theta_0)$ is exact and still forecasts nothing — using it for $n$ steps requires $n$ binary digits of the initial condition, the exponential precision demand that IS the positive Lyapunov exponent. Determinism, solvability, and predictability are three independent properties; the logistic map at $r = 4$ has the first two and not the third, and internalizing that triage is the final exam of this chapter.

Worked examples

1 · An ecology reading of the multiplier (confidence builder)

A fishery's stock is modelled by the logistic map with $r = 2.6$ . Find the equilibrium, classify the approach, and estimate how many seasons a 10% overfishing shock takes to decay below 1%.

Step 1. Equilibrium: $x^* = 1 - 1/2.6 = \boxed{0.615}$ of carrying capacity.

Step 2. Multiplier: $\mu = 2 - r = -0.6$ . Stable ( $|\mu| < 1$ ), and alternating: the stock overshoots and undershoots equilibrium in alternate seasons as it recovers — damped population oscillation with no external driver, purely from the overshoot dynamics of discrete generations.

Step 3. Decay: deviations shrink by $|\mu| = 0.6$ per season; need $0.6^n < 0.1$ , so $n > \ln 10/\ln(1/0.6) = 2.303/0.511 = 4.5$ — about $\boxed{5}$ seasons, flipping sign each season on the way in. (At $r = 3.2$ the same computation gives $|\mu| = 1.2 > 1$ : the shock grows, and the stock settles into the 2-cycle of the cascade article — boom-bust as an attractor, not an accident.)

2 · Using the exact chaotic solution (exam level)

For $r = 4$ with $x_0 = 0.3$ : (i) compute $x_5$ from the closed form and check against direct iteration; (ii) determine how many decimal digits of $x_0$ are needed to predict $x_{50}$ to within 0.01; (iii) compute the exact fraction of time the orbit spends in $[0.9, 1.0]$ .

Step 1. Angle: $\theta_0 = \tfrac1\pi\arcsin\sqrt{0.3} = 0.184505$ . Closed form: $x_5 = \sin^2(2^5\pi\theta_0) = \sin^2(32\pi \times 0.184505) = \boxed{0.08795}$ . Direct iteration $0.3 \to 0.84 \to 0.5376 \to 0.99435 \to 0.02249 \to 0.08795$ ✓ — agreement to five digits (and to 13 in double precision).

Step 2. Digit budget: each iteration consumes one binary digit of $\theta_0$ ; predicting $x_{50}$ needs roughly $50 + \log_2(1/0.01) \approx 57$ bits, i.e. $57 \times 0.301 = \boxed{17}$ decimal digits of the initial condition — beyond any physical measurement, in a system whose solution is a one-line formula. Solvable, deterministic, unpredictable.

Step 3. Time in $[0.9, 1]$ : ergodicity converts the question to the invariant density, $\int_{0.9}^{1}\frac{\mathrm{d}x}{\pi\sqrt{x(1-x)}}$ . With $\int\rho = \tfrac{2}{\pi}\arcsin\sqrt{x}$ : fraction $= \tfrac2\pi[\arcsin(1) - \arcsin\sqrt{0.9}] = \tfrac2\pi[1.5708 - 1.2490] = \boxed{0.205}$ — over a fifth of all iterations land in the top decile, the arcsine law's endpoint-lingering made quantitative (and visible as Figure 3's right-hand spike).

3 · Dating the period-3 window (applied)

(i) At what $r$ does the period-3 window open, exactly? (ii) Just below it, at $r = 3.8280$ , estimate the character of the dynamics. (iii) Inside the window, where does the 3-cycle's own chaos arrive, roughly?

Step 1. Exact opening (Derivation C): $r_3 = 1 + \sqrt8 = \boxed{3.828427}$ — a tangent bifurcation of $f^{(3)}$ , stable and unstable 3-cycles born together.

Step 2. At $r = 3.8280$ , the gap is $r_3 - r = 4.3\times10^{-4}$ . The saddle-node ghost gives laminar period-3 episodes of typical length $T \sim c\,(r_3 - r)^{-1/2} \approx c \times 48$ iterations (with $c$ of order unity) separated by chaotic bursts: intermittency — a signal that looks periodic for dozens of generations, then scrambles, then locks again. Time series inspection would tempt a false "period 3" diagnosis; the $\lambda(r)$ curve stays (barely) positive.

Step 3. Inside the window the stable 3-cycle runs its own Feigenbaum cascade. Its width is set by the same universal machinery: the window spans roughly $3.8284$ to $3.8415$ (closing crisis), with the internal $3\cdot2^n$ cascade accumulating around $r \approx 3.8496$ 's neighbourhood scaled into the window — practically, chaos-within-the-window arrives after an interval $\sim(1/\delta)$ -scaled copies of the doubling ladder, a miniature of Figure 2 nested inside itself. The bifurcation diagram is self-similar not only at the cascade but around every window: Sharkovskii's zoo, rendered fractally.

Origins · 1 minRobert May and how this was discovered

Robert May

1936–2020 · Sydney · Princeton · Oxford

May appears twice in this chapter's cast — the bifurcations-overview article introduced him — and this closing article is the natural home of his masterpiece. Trained as a theoretical physicist in Sydney (plasma physics), he defected to ecology at Princeton and brought with him a physicist's instinct for minimal models. Working through the discrete logistic equation as a model of insect generations, he found — before the vocabulary existed to say it — that the model's failures to settle were not pathologies but the point: deterministic populations could oscillate and turn effectively random with no environmental noise whatsoever. His 1976 Nature review made the one-line map the founding document of applied chaos, and its closing "evangelical plea" — that students meet this equation early, to learn how simple rules breed complex behaviour — reshaped curricula across sciences. May's later career kept the theme at ever-larger scale: chaos and stability in ecosystems (his paradox that complexity does not imply stability), epidemic dynamics with Anderson (the $R_0$ framework that governed twenty-first-century pandemic policy descends directly from it), and systemic risk in banking networks after 2008. President of the Royal Society, chief scientific adviser to the UK government, a life peer — Baron May of Oxford, the only member of this chapter's gallery to sit in a legislature. The parabola was the smallest thing he studied, and the biggest.

History

1845

— Verhulst formulates the continuous logistic equation for saturating population growth; the map is its discrete-generation shadow.

1964

— Sharkovskii publishes the total ordering of periods for interval maps; period three sits at the summit, unnoticed in the West for a decade.

1975

— Li and Yorke prove period three implies chaos, coining the word chaos in its modern mathematical sense.

1976

— May's Nature review makes the logistic map the universal teaching laboratory of nonlinear dynamics.

1978

— Feigenbaum's universality places the map's cascade at the centre of renormalization physics, completing its promotion from ecology sketch to fundamental object.

Going further

References. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., ch. 10 (one-dimensional maps) · May, Nature 261, 459 (1976) · Ott, Chaos in Dynamical Systems, 2nd ed., ch. 2 · Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., ch. 11 · Li & Yorke, Amer. Math. Monthly 82, 985 (1975) · Feigenbaum, J. Stat. Phys. 19, 25 (1978) Key numerical landmarks: fixed point $x^* = 1 - 1/r$ with multiplier $\mu = 2 - r$ , transcritical exchange at $r = 1$ , superstability at $r = 2$ . Cascade values $r_1 = 3$ , $r_2 = 1+\sqrt6$ , $r_\infty = 3.569946$ , $\delta = 4.669201$ . The $r = 4$ conjugacy to the doubling map yields the closed-form solution, $\lambda = \ln 2$ , and the arcsine density; Worked Example 2's $x_5 = 0.08795$ was recomputed directly. The period-3 window opens at the exact value $r = 1 + \sqrt8 = 3.828427$ ; Jakobson's theorem (1981) establishes that chaotic parameters form a positive-measure set.

Builds on: Fixed Points and Equilibria · Linear Stability Analysis and the Jacobian · Bifurcations and the Loss of Stability Leads to: Period-Doubling Cascade and the Route to Chaos · Saddle-Node, Transcritical, and Pitchfork Bifurcations · Phase Portraits and the Phase Plane

Is the logistic map actually a good population model, or just a good chaos demo?

As quantitative ecology, it is a caricature — real populations have age structure, stochasticity, spatial spread, and rarely the precise parabolic density dependence. As conceptual ecology, it was revolutionary and remains honest: it proves that erratic population data need not imply environmental noise or measurement error, because the simplest possible self-limiting growth rule generates boom-bust cycles and aperiodicity endogenously. That reframing — "before fitting noise, check whether your deterministic skeleton oscillates" — is May's permanent contribution to the field. Working fisheries and pest models use better-behaved cousins (the Ricker map $x e^{r(1-x)}$ avoids the negative-population pathology at high growth; Beverton–Holt is the tamer contest-competition analogue), but these are unimodal maps in the same universality class: cascades, windows, Feigenbaum spacing and all. So the demo IS the model class; only the parabola's specific coefficients are cartoonish.

What exactly happens for r just above 4?

The attractor dies and a chaotic repeller is born — the map's third life. For $r > 4$ the parabola's peak $r/4$ pokes above 1, so a central sub-interval of points maps above 1 and then escapes to $-\infty$ : almost every initial condition eventually leaks out. What remains forever is the set of points whose entire forward orbit avoids the escape gap — and removing the gap's preimages generation by generation is precisely a middle-gap Cantor construction: the invariant set is a fractal dust (for $r$ slightly above 4, dimension just below 1) on which the dynamics is fully chaotic but non-attracting. Orbits near it perform transient chaos: wander erratically for a while, then abruptly escape — with exponentially distributed lifetimes governed by an escape rate. This is the simplest model of chaotic scattering, the bounded-time cousin of chaos that governs, e.g., how comets transit planetary systems — and a reminder that invariant chaotic sets matter even when nothing settles onto them.

The bifurcation diagram shows dark bands merging as r decreases toward the accumulation point. What are they?

The mirror-image staircase of the cascade — the band-merging or inverse cascade. Just above $r_\infty$ , the chaotic attractor is not one interval but $2^n$ narrow bands, visited in strict cyclic order (the ghost of the period- $2^n$ orbit whose doubling created them); within each band the visits are chaotic. As $r$ increases the bands widen and merge in pairs — $2^n$ bands to $2^{n-1}$ — at parameter values that accumulate geometrically at $r_\infty$ from above with the same Feigenbaum ratio $\delta$ : the diagram is self-similar on both sides of the critical point, exactly like correlation structure on both sides of a phase transition. The mergings are homoclinic events (bands touch the unstable orbit between them), and observationally they matter: a period- $2^n$ -banded chaotic signal still shows sharp subharmonic spectral lines at $f/2^n$ atop broadband noise, so a spectrum can look "almost period-8" while being genuinely chaotic. Counting band mergings is one of the standard ways experiments locate $r_\infty$ from the chaotic side.

Does the logistic map have chaos on a set of parameters you could actually hit, or do windows fill everything?

A genuinely deep question with a famous answer. The windows are dense: arbitrarily close to any parameter in $(r_\infty, 4)$ there is an interval of $r$ with a stable periodic orbit — so "chaotic $r$ " contains no interval at all, and one might fear chaos is a measure-zero mirage. Jakobson's theorem (1981) resolves it the other way: the set of parameters for which the logistic map has an absolutely continuous invariant measure — honest, observable chaos with positive exponent — has positive Lebesgue measure. Later work (Lyubich, Graczyk–Świątek) completed the astonishing dichotomy: almost every parameter in $[0,4]$ is either regular (attracting periodic orbit) or stochastic (chaotic with a.c.i.m.), with the two sets interleaved on all scales — chaos and order form a fat fractal partition. Practically: pick $r$ "at random" in the chaotic zone and you land on genuine chaos with positive probability, but you are always within $\varepsilon$ of a window; numerically, finite precision and finite observation time blur the finest windows into the chaotic background, which is why Figure 2 looks the way it does.

How literally does a Poincaré section of a real flow reduce to a one-dimensional map like this?

Quantitatively, and the driven pendulum already showed the mechanism: its stroboscopic map contracts areas by $\det DP = e^{-2\beta T} \approx 0.009$ per return, so the 2D section dynamics collapses, within a couple of returns, onto a nearly one-dimensional curved arc — and the induced dynamics along that arc is a unimodal map with a quadratic fold (the fold coming from the flow's stretch-and-fold projected onto the arc). That is why the pendulum's cascade obeys the logistic map's constants. The reduction is an approximation with a controlled error: the invariant arc has tiny transverse thickness $\sim \det DP$ , so the 1D map description misses effects at that scale (e.g. the fine splitting of the attractor's Cantor layers). The same collapse happens in the Lorenz system, historically first: Lorenz plotted successive maxima of $z(t)$ against each other and found the points falling on a sharp tent-like curve — the "Lorenz map" — whose one-dimensional analysis correctly predicts the flow's chaos. The moral runs both directions: 1D maps inherit their physical legitimacy from dissipative flows, and dissipative flows inherit their universal numbers from 1D maps.

If period three implies chaos, and the period-3 window contains a STABLE 3-cycle, is the map in the window chaotic or not?

Both, in the two different senses that this chapter has been careful to separate. Li–Yorke chaos is a statement about the existence of complicated orbits: once a period-3 orbit exists (anywhere at or beyond the window's birth), the map possesses periodic orbits of every period and an uncountable scrambled set of aperiodic orbits. All of that genuinely exists inside the window — but it is all unstable, an invisible skeleton of measure zero. Observable, attractor-sense chaos asks a different question: where do typical orbits go? Inside the window, almost every initial condition converges to the stable 3-cycle, the observed Lyapunov exponent is negative, and measurements show clean period-3 — no observable chaos. The unstable skeleton is not idle, though: it produces long chaotic transients (orbits rattle around the ghost of the former chaotic set before locking onto the 3-cycle), and it is the raw material for the crisis that ends the window, when the re-expanding chaotic set collides with the unstable 3-cycle and observable chaos returns. So: Li–Yorke chaotic throughout, observably chaotic outside the stable windows — the distinction between existence and visibility, one last time.

Discrete Maps and the Logistic Map

Where this lives today

The physical picture

The map on its own terms

Which tool answers which question

Limiting cases

Worked examples

1 · An ecology reading of the multiplier (confidence builder)

2 · Using the exact chaotic solution (exam level)

3 · Dating the period-3 window (applied)

Going further

Before this

Where this leads