Chaos and Fixed Points

What iteration does: convergence, stability and its loss, period doubling, bounded chaos, self-similar sets, and the fixed points behind self-reference.

Thirteen simulations about what happens when you apply the same rule over and over. The first few converge. Then convergence fails in an orderly way, and the orderly failure turns out to have universal structure. The last four are the same idea in logic and type theory, where a fixed point of self-application is what Gödel, Turing, and Cantor all turn on. Everything here is deterministic: no randomness is involved anywhere on this page.

Contraction to a fixed point

What makes a repeated process settle down?

Several starting values collapse toward the same fixed point. The distance to x* = 1 halves at every step.

Try this. Start the runs from wildly different places. They arrive at the same point, and the distance falls by the same factor each step regardless of where you began.

Assumptions and limits. This is the Banach fixed-point theorem with a particular map, and it works because that map is a contraction. Most iterations are not, and nothing here tells you whether a given one is.

Essay: Fixed Points and Attractors · Source

Reading iteration off a graph

How do you see the future of an iteration without computing it?

A cobweb diagram draws iteration directly: move vertically to y = f(x), horizontally to y = x, and repeat.

Try this. Follow one step at a time and watch the staircase form. Where the curve crosses the diagonal shallowly the path spirals in; where it crosses steeply it spirals out. The slope at the crossing is the whole story.

Assumptions and limits. Only works for one-dimensional deterministic maps. Higher-dimensional systems have no equivalent picture, which is part of why they are harder to reason about.

Essay: Fixed Points and Attractors · Source

Losing stability

Can a fixed point still exist after it stops mattering?

Move r. The fixed point still exists after r = 3, but |f'(x*)| crosses 1 and the point stops attracting nearby states.

Try this. Move r through 3 slowly. The fixed point does not disappear — it is still a solution — but nearby states stop being drawn to it, which makes it unobservable in practice.

Assumptions and limits. The derivative test is local. It tells you what happens to states already near the fixed point and nothing about behaviour that starts far away.

Essay: Fixed Points and Attractors · Source

The logistic map

How does a rule with no randomness in it produce something that looks random?

The logistic map's long-run states split from one branch to two, four, eight, and then a chaotic cloud with periodic windows.

Try this. Sweep r upward and count the branches: one, two, four, eight, and then a cloud. Then look inside the cloud for the periodic windows, where order returns without warning.

Assumptions and limits. Fully deterministic. The irregularity is sensitivity to initial conditions, not noise — the same start always gives the same path. That distinction matters, because real data has both.

Essay: Chaos, Fractals, and Renormalization · Source

Feigenbaum’s constant

Is the route into chaos the same in systems that have nothing to do with each other?

Successive period-doubling intervals shrink toward Feigenbaum's universal constant delta ≈ 4.669.

Try this. Measure the spacing between successive doublings and take the ratios. They approach 4.669 — a number that appears in dripping taps and convecting fluids as well as in this map.

Assumptions and limits. Universality holds for a class: unimodal maps with a quadratic maximum. It is a real theorem about a real class, not a claim that every system approaching chaos does this.

Essay: Chaos, Fractals, and Renormalization · Source

Stretch and fold

What is chaos, geometrically?

The tent map stretches nearby points apart and folds them back into the interval. That is the basic geometry of bounded chaos.

Try this. Start two points a hair apart and step forward. Stretching separates them; folding keeps them inside the interval. Divergence without escape is the entire mechanism.

Assumptions and limits. Piecewise-linear and idealized, which is what makes the mechanism visible. Smooth systems do the same thing less legibly.

Essay: Chaos, Fractals, and Renormalization · Source

The Cantor set

What does structure at every scale actually look like?

Move the stage count. The Cantor set is made by repeating the same removal rule, leaving two scaled copies at every stage.

Try this. Advance the stages one at a time. Each stage is two shrunken copies of the previous one — the set is a fixed point of its own construction rule.

Assumptions and limits. The real Cantor set is the limit of infinitely many stages. The screen shows a finite approximation, and pixels put a floor on how far the self-similarity can be drawn.

Essay: Chaos, Fractals, and Renormalization · Source

One basin becomes two

How does a system with one stable state end up with two?

Move the developmental parameter. One basin becomes two, which is the simplest mathematical picture of a progenitor state splitting into committed fates.

Try this. Move the parameter through the split. Before it, there is one place to end up and the choice does not exist. After it, the old state is a ridge and the outcome depends on which side a small nudge lands.

Assumptions and limits. Waddington’s landscape was a metaphor; this makes it concrete as a pitchfork bifurcation in one variable. Real differentiation involves many interacting genes and is not one-dimensional.

Essay: Life as a Double Fixed Point · Source

Continued fractions as dynamics

What does approximating a number have to do with chaos?

Move the initial number and replay the orbit. The Gauss map G(x) = {1/x} generates continued-fraction digits, showing that rational approximation is itself driven by a chaotic dynamical system.

Try this. Change the starting number slightly and compare the digit sequences. They diverge, which is the point: the arithmetic of approximation is a chaotic orbit.

Assumptions and limits. Floating point. After enough steps the computed orbit has left the true one entirely — an honest demonstration of sensitivity, and a reason not to read late digits as exact.

Essay: Number Theory and the Arithmetic of Recurrence · Source

Small divisors

Which numbers are hardest to approximate with fractions, and why does it matter?

Small values of |qω - p| are near-resonances. The golden ratio avoids exceptionally small denominators better than many other irrationals.

Try this. Compare the golden ratio against a number with a large continued fraction coefficient. The near-resonances the second one hits are the small divisors that wreck perturbation series.

Assumptions and limits. An illustration of Diophantine approximation over a finite range of denominators, not a proof of any result about it.

Essay: Number Theory and the Arithmetic of Recurrence · Source

Data types as fixed points

In what sense is a list a solution to an equation?

A finite list is built by repeatedly unfolding List(A) = 1 + A x List(A) until the empty case stops the recursion.

Try this. Unfold step by step. Each unfolding substitutes the definition into itself; the empty case is what makes the process terminate rather than run forever.

Assumptions and limits. A schematic of an initial algebra. It ignores strictness, laziness, and the question of which such equations have solutions — all of which a real type system has to answer.

Essay: Logic, Self-Reference, and Category Theory · Source

The diagonal argument

Where does self-reference come from?

A self-referential fixed point appears when a representation is fed back into the rule that interprets it.

Try this. Follow the construction: a representation goes into the rule that interprets representations, and the result refers to itself. Nothing is smuggled in — the self-reference is produced by the feedback.

Assumptions and limits. The skeleton of the argument, not a formal proof. Gödel’s version requires arithmetization, which is the hard part and is not shown here.

Essay: Logic, Self-Reference, and Category Theory · Source

Lawvere’s fixed-point theorem

Is there one theorem underneath Gödel, Turing, and Cantor?

Replay the categorical skeleton: representation gives evaluation, evaluation gives the diagonal, and the diagonal either forces a fixed point or proves that complete self-representation was impossible.

Try this. Replay the chain and note where it forks. Either every endomap has a fixed point, or the representation you assumed cannot exist. The famous impossibility results are the second branch.

Assumptions and limits. Assumes a cartesian closed category. Getting from the abstract statement to any specific incompleteness theorem takes real work that this replay skips.

Essay: Logic, Self-Reference, and Category Theory · Source