The collection

Every essay — page 28

Page 28 of 40, continuing through the fields in the same order.

Geometry Analysis Algebra Discrete Topology Probability Number Dynamics Logic Computation Applied What's new Series Concepts Search

Dynamics

One rule, applied over and over, and what the sequence does in the end.

67 starting points that find all 5 roots. The roots of z⁵ − 1 with a ring of starting points around them, each start marked by which root the method reaches from it, and every root reached by at least one.

Covering rather than avoiding

The turning point's orbit over the bifurcation diagram, 3.55 to 4. The bifurcation diagram of the logistic map from 3.55 to 4 with the curves f(1/2), f²(1/2), … up to the 6th image drawn over it. The first two bound the attractor and the rest trace the dark lines inside it.

The dark lines are one point's orbit

Intermittency at r = 1 + √8 − 0.0003. A time series of 600 steps of the logistic map just below the period-three window. Long stretches that look like a cycle of three, shaded, alternate with irregular bursts; there are 6 such stretches here.

The window that opens with a stutter

Rule 184 at density 0.30. A space-time diagram of rule 184 on a ring of 120 cells, 80 steps down the page, starting from a random row with 36 cars. The diagonal stripes are free-moving cars; the jams dissolve.

A road where nobody overtakes

A local rule taking a vote. A space-time diagram of the GKL rule on 149 cells from a random row with 69 ones. Black and white regions grow and meet along slanting boundaries, and after 69 steps the whole ring is 0.

No local rule can count the votes

6 figures $How long four orbits of the doubling map spend in each quarter. Grouped bars for four starting points of the doubling map, each group showing the fraction of time the orbit spends in each quarter of the unit interval, against a dashed line at one quarter.$

Almost every orbit is fair

The Chebyshev maps T₂, T₃, T₄ and T₅. Four small square plots of the Chebyshev polynomials of degrees two to five on the interval from minus one to one, each with the diagonal drawn; the graph of degree n sweeps between the bottom and top of the square n times.

A solvable chaos of every degree

A periodic point and a wandering one, both near 0.3, parted by step 4. The distance between the orbit of a periodic point and the orbit of a point from a dense orbit, both starting in the same small interval, plotted against the step until the wandering orbit nears the point farthest from the periodic one.

Sensitivity comes free

What the tent of slope 3 keeps: 32 pieces after 5 steps. Rows showing the parts of the unit interval that remain inside it for 0 to 5 steps of the open tent map of slope 3, halving into a Cantor set.

Chaos on a set nobody lands on

How many steps Newton's method needs on (z − 1)²(z + 1). A square of the complex plane shaded in bands by the number of iterations a root-finding method needs from each starting point, darker meaning slower, with the roots marked.

A double root halves the error instead of squaring it

Newton's and Halley's basins for z³ − 1. Two squares of the complex plane side by side, each coloured by which root a starting point converges to, the left under Newton's method and the right under Halley's, with non-converging starts marked.

A cubic method that is Newton's in disguise

Every orbit round a square closes. 5 outer-billiard orbits about a square, each drawn as its closed ring of points, with periods 4, 8, 12, 20, 24 growing outwards.

The ball that stays outside the table

A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH.

The word a straight line spells

Two staircases and the interval between them, at K = 1.5. A plot against the drive of the upper and lower ends of the circle map's rotation interval above the critical line, two stepped curves with the band between them shaded.

A whole interval of speeds

The standard map as its kick grows. Three square phase portraits of the standard map at increasing strengths, dotted with orbits: curves spanning the square at the smallest, fewer at the critical value, and a scattered sea with islands at the largest.

The last circle to break

Writing 0.3 in base φ. The graph of x ↦ βx mod 1 for the golden ratio φ, the diagonal, and the cobweb staircase of the orbit of 0.3, beside the 8 digits it writes.

Counting in a base that is not a whole number

Where x ↦ βx mod 1 spends its time, the golden ratio φ. A histogram of 2000000 orbit points of the β-map for the golden ratio φ in 50 bins, with Parry's staircase density drawn over it; they agree, and the density steps down at the orbit of 1.

Where a base-β orbit spends its time

The Collatz map on the 2-adic integers, before and after changing to parity coordinates. Two scatter plots of 2048 points: the Collatz map on 2-adic integers in binary-digit coordinates, a scattered cloud, and the same map in parity-vector coordinates, where every point lies on the two lines of the doubling map.

Where the Collatz map is a coin

5 figures $PRIMEGAME: fourteen fractions whose powers of two are the primes. The size in binary digits of the numbers produced by Conway's PRIMEGAME from 2 over 40000 steps, with the pure powers of two marked; their exponents are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.$

Fourteen fractions that list the primes

Where one Hénon orbit spends a million steps. A density map of 1000000 steps of one Hénon orbit counted into 420 × 140 cells: 2542 cells visited, half the time spent in 608 of them.

Where the time goes on an attractor

A quadrilateral the Hénon map carries strictly inside itself. Hénon's trapping quadrilateral with corners (-1.33, 0.42), (1.32, 0.133), (1.245, -0.14), (-1.06, -0.5), the image of its boundary under the Hénon map lying strictly inside it, and an orbit on the attractor.

A region the orbit cannot leave

A strand of the logistic map's diagram between r = 3.8284 and 3.8572, upside down. The bifurcation diagram of the logistic map cropped to r in [3.8284, 3.8572] and x in [0.44, 0.565], with the value axis inverted.

Every window is the whole diagram again

How many windows of each period, on two maps and as a count of necklaces. period 1: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 2: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 3: 1 (logistic), 1 (sine), 1 (necklaces), 1 (polynomials); period 4: 2 (logistic), 2 (sine), 2 (necklaces), 2 (polynomials); period 5: 3 (logistic), 3 (sine), 3 (necklaces), 3 (polynomials); period 6: 5 (logistic), 5 (sine), 5 (necklaces), 5 (polynomials); period 7: 9 (logistic), 9 (sine), 9 (necklaces), 9 (polynomials); period 8: 16 (logistic), 16 (sine), 16 (necklaces), 16 (polynomials); period 9: 28 (logistic), 28 (sine), 28 (necklaces), 28 (polynomials); period 10: — (logistic), — (sine), 51 (necklaces), 51 (polynomials); period 11: — (logistic), — (sine), 93 (necklaces), 93 (polynomials); period 12: — (logistic), — (sine), 170 (necklaces), 170 (polynomials); period 13: — (logistic), — (sine), 315 (necklaces), — (polynomials); period 14: — (logistic), — (sine), 585 (necklaces), — (polynomials).

Windows counted like necklaces

Traffic with random dawdling: jams from nowhere at density 0.18. Nagel–Schreckenberg traffic, 29 cars on 160 cells, top speed 5, dawdling probability 0.25; up to 11 cars stopped at once.

A jam that comes from nowhere