Deterministic chaos

A precise rule can have surprising outcomes

Take a number between 0 and 1. If it is no larger than 1/2, double it. If it is larger than 1/2, subtract twice the number from 2. Repeat the same rule on each new result. Starting at 5/8 gives 3/4, then 1/2, then 1, then 0. After that the number remains 0.

There is no coin toss in this example. Given an exact starting value, the rule determines every subsequent value. Deterministic means that the next state is fixed by the current state and rule; it does not promise that approximate starting information remains adequate far into the future.

The values are normalized, dimensionless states in a mathematical model. They are not measured positions or velocities. We will use this simple rule to distinguish an exact prediction, a comparison of nearby starts and a claim about a whole system.

Substitution and graph reading are the main preparation; Algebra, functions, graphs, and logarithms provides a return path. Follow the rule and exact table first. The interval argument later is an optional deeper check; the branch-slope discussion can be revisited with Derivatives, Taylor series, and local approximation.

Define the state, interval and update times

Call the current state x and the next-state rule F. This standard tent map is F(x) = 2x for 0 ≤ x ≤ 1/2, and F(x) = 2 − 2x for 1/2 < x ≤ 1. The complete state space is the real interval [0, 1]. Both formulas give 1 at the meeting point, so F is continuous and every output stays in the interval.

Write x_n for the state after n updates, where n = 0, 1, 2, … and x_0 is the supplied starting state. The evolution rule is x_(n+1) = F(x_n). For this teaching case assign updates one second apart, t_n = n × 1 s. The index n is a count; t_n has units of time.

No state between these update times is supplied. This is a discrete map, not a continuous flow or a differential equation. Joining its successive states by a smooth curve would add an assumption about the intervening motion.

On the two open branches, the input-output slopes are 2 and −2. These are dimensionless gains, not rates in s⁻¹. At x = 1/2 the one-sided slopes disagree, so the ordinary two-sided derivative does not exist even though the rule itself is continuous.

A fixed point satisfies F(x) = x. The left branch gives x = 0 and the right gives x = 2/3. The dashed identity line in the upper graph makes these intersections visible.

Watch separation grow, then vanish

Keep a reference sequence at a_0 = 2/3. It remains a_n = 2/3 because 2 − 2(2/3) = 2/3. Start a second sequence at b_0 = 2/3 + 1/1536 = 1025/1536. Define the signed difference δ_n = b_n − a_n and the nonnegative separation d_n = |δ_n|.

For the updates from states n = 0 through n = 8, both current states lie on the right branch. Subtracting their update equations gives δ_(n+1) = −2δ_n. Thus δ_n = (−2)^n/1536 through state n = 9: the sign alternates while its magnitude doubles.

At state n = 9 the second sequence is b_9 = 1/3, while the reference is still 2/3. The next update uses different branches: F(1/3) = F(2/3) = 2/3. They merge at n = 10 and remain together thereafter. The earlier doubling does not continue through that fold.

The upper graph plots input against next output, not state against time. The lower panel shows unconnected samples d_n at integer update indices. Its first positive value is too small to resolve from the zero axis by marker position alone; the exact table distinguishes it from the zero at n = 10.

This selected pair is eventually periodic. It illustrates finite separation and merging; it does not establish an aperiodic trajectory or prove that the whole map is chaotic.

An error bound is a conditional guarantee

For any two inputs in [0, 1], |F(x) − F(y)| ≤ 2|x − y|. On the same straight branch there is equality. Across the corner, with x ≤ 1/2 ≤ y, the output difference is 2|1 − x − y|, which is no larger than 2(y − x). This bound covers the fold as well as both branches.

If the initial separation is at most ε, repeated use of this bound gives d_n ≤ 2^n ε. Since both states also lie in [0, 1], d_n ≤ min(1, 2^n ε). Here ε is a deterministic bound on initial-state difference, not a standard deviation or a probability statement.

For ε = 1/1536 and a required separation no larger than 1/100, the bound guarantees that tolerance through n = 3: 2^3/1536 = 1/192 < 1/100. At n = 4 it gives 1/96 > 1/100, so this bound no longer supplies the guarantee. The selected pair actually exceeds the tolerance there.

A failed sufficient bound does not require every pair to fail. At n = 10 this pair has separation zero, although the upper bound is still 2/3. A bound describes what can be guaranteed under its assumptions, not the exact value of every particular comparison.

A chaotic map is a claim about the whole interval

Chaos has several mathematical definitions. Here we use a Devaney-style criterion for a continuous map of an interval into itself: topological transitivity, a dense set of periodic points and sensitive dependence on initial states. All three descriptions refer to the declared complete state space, not only to the selected sequence.

Topological transitivity means that for any two nonempty open regions U and V within [0, 1], some positive number of updates takes at least one point of U into V. An open region contains a neighborhood within the state interval; at an endpoint, only its part inside [0, 1] is relevant.

A periodic point returns after a positive number of updates. Periodic points are dense if every nonempty open interval within the state space contains one. Density allows many different periods nearby; it does not say that every point is periodic.

Sensitive dependence requires a fixed positive separation threshold. For every initial x and every positive closeness δ, there must be another state y in [0, 1] with 0 < |y − x| < δ whose later separation from x reaches that threshold. The nearby state and the later update can depend on x and δ. This is not a requirement that every nearby pair keep separating.

Why the tent map meets this criterion

Write F^k for applying F exactly k times, not for raising one output to a power. After k ≥ 1 updates, the interval splits into 2^k closed pieces J_(k,j) = [j/2^k, (j + 1)/2^k], where j = 0, …, 2^k − 1. On each piece, F^k is a straight branch covering the whole output interval [0, 1]. Neighboring formulas agree at shared endpoints.

For even j the branch formula is F^k(x) = 2^k x − j; for odd j it is F^k(x) = j + 1 − 2^k x. At k = 1 these are the original two branches. Each full branch has one input mapped to 1/2; cutting there and applying F makes two new full branches. This gives the claim at the next update and establishes it for every positive k.

These pieces have width 2^(−k), so any nonempty open interval U contains a whole sufficiently small piece. Its image under F^k is [0, 1] and therefore meets every destination region V. Since F maps the whole interval onto itself, the same remains true for every later update. This proves transitivity and the stronger property called mixing.

On each full piece, the continuous function F^k(x) − x has opposite or zero signs at its endpoints. It must therefore vanish somewhere between them. That point returns after k updates, although its least period may be smaller. A full piece can be placed inside any open interval, so periodic points are dense.

For sensitivity, take any initial x and any positive δ. Place a full piece inside its δ-neighborhood within [0, 1]. Its k-th image reaches both 0 and 1. At least one of these endpoints lies at distance at least 1/2 from F^k(x). Its preimage y in the chosen piece is arbitrarily close to x and separates by at least 1/2 at update k. The single threshold 1/3 therefore works for every x and neighborhood.

The complete real-interval map meets the stated criterion. The proof comes from its full branch structure, not from the length of our table, a visually irregular trace or the slope at one selected point.

Exceptional paths and finite representations still matter

The fixed states 0 and 2/3 coexist with the chaotic interval map. Another exact path is 2/5 → 4/5 → 2/5, with least period two. The system-level criterion permits such exceptional periodic trajectories.

For an exact rational input a/b with positive integer b, the affine rule can retain the same denominator b at every update. Only the b + 1 integer numerators from 0 through b are possible. A repeated state must eventually occur and the future then repeats. This does not force every rational input to reach 0: the period-two example remains nonzero.

A dyadic input is a number k/2^p with integer p ≥ 0 and 0 ≤ k ≤ 2^p. Each update removes at least one denominator power of two until the state is 0 or 1; one further update reaches 0. At most p + 1 updates suffice. The opening path 5/8 → 3/4 → 1/2 → 1 → 0 is an exact example.

An exact irrational input stays irrational under either branch. Any periodic point must solve an affine equation with integer intercept and slope 2^k or −2^k, so it is rational. Thus an irrational start such as √2/2 is never eventually periodic in this exact map. That fact alone does not prove that its orbit is dense or statistically typical.

A fixed finite set of computer states with a deterministic update that stays in the set eventually repeats too. Rounded computation is not automatically the exact real-interval map. Floating-point arithmetic and numerical stability provides a return path for this distinction; no particular implementation or measured numerical error is assumed here.

What can a real record establish?

The tent map meets a precise chaos criterion while using a completely deterministic rule. Randomness is not needed for this mathematical example. Conversely, an irregular or noisy measurement record does not by itself prove deterministic chaos.

A physical claim needs identified state variables and units, a justified evolution model, measurement-error analysis and evidence that the conditions under study are represented. Our normalized interval rule and assigned update spacing are supplied teaching choices, not observations of a device or a planet.

A finite trace can test specified finite predictions. It cannot alone establish the interval-wide and indefinite-time properties proved above. Keep the complete model, exact initial information, approximate input bounds and finite computed output distinct.

Lyapunov exponents and predictability examines growth-rate ideas further, and Iterated maps and fractal structures develops repeated-map geometry. They are further destinations, not preparation required to apply and check this lesson’s supplied rule.

Worked examples

Try it yourself

1. Starting at 2/5, calculate two updates. What is the least period, and does that periodic path refute the whole-interval chaos criterion?

   Show the answer and reasoning

   The path is 2/5 → 4/5 → 2/5, with least period two. It does not refute the interval-wide claim.

   The first input uses the doubling branch and the second uses 2 − 2x. The states differ, so the least period is not one. The stated chaos criterion permits, and requires a dense set of, periodic points.

2. Starting at 5/8, calculate four updates. Must every rational starting state reach zero?

   Show the answer and reasoning

   The updates give 3/4, 1/2, 1 and 0. Every rational start need not reach zero.

   The supplied dyadic denominator is 2^3, so the stated p + 1 bound allows four updates to reach zero. The exact rational path 2/5 ↔ 4/5 supplies a nonzero periodic counterexample to the broader claim.

3. Find F(1/2). Is the ordinary derivative there 2 because the value uses the left branch? Is a branch gain a velocity?

   Show the answer and reasoning

   F(1/2) = 1. The ordinary derivative does not exist there, and a dimensionless branch gain is not a velocity.

   The branch-assignment convention fixes the value but does not remove the two-sided derivative requirement. Nearby left and right quotients have slopes 2 and −2. No continuous-time motion has been supplied.

4. The first lower-panel circle appears almost on the zero axis. What is its exact separation, and may its pixel position alone establish a zero?

   Show the answer and reasoning

   The initial separation is 1/1536, which is positive. The marker position alone cannot establish zero.

   At 900 px per separation unit this value lies only 75/128 px above the baseline, smaller than the marker size. The exact table records the positive value; the square at n = 10 separately marks the actual zero.

5. Our two selected starts merge at n = 10, and the reference start stays fixed. Does sensitive dependence require every nearby pair to keep separating? Does the selected trace alone prove the criterion?

   Show the answer and reasoning

   No to both. Sensitive dependence is an existence statement for each base state and neighborhood; the selected trace alone is insufficient.

   There must be some arbitrarily nearby start that reaches a common positive separation threshold at some later update. It need not be this selected start, and separation need not persist forever. The complete interval argument, not the finite comparison, establishes the three stated system-level properties.

6. In exact real arithmetic, can the starting state √2/2 become eventually periodic under this tent map? Does this conclusion automatically describe its rounded computer representation?

   Show the answer and reasoning

   It cannot become eventually periodic in the exact model. A rounded representation is a different input and computation.

   Both affine branches preserve irrationality. A point fixed by F^k solves an affine equation with integer intercept and slope ±2^k, so that point is rational. An eventually periodic irrational would have to become such a rational point, which is impossible. No density or typicality claim follows from this reasoning.

What to carry forward

• A discrete deterministic rule assigns a unique next state; it need not supply a continuous flow or unique inverse.
• The selected exact pair separates for a finite interval and then merges under the same rule.
• A system-level chaos criterion is stronger than a finite divergent or irregular trace.
• The tent map’s full interval structure coexists with exceptional fixed, periodic and eventually periodic trajectories.
• Conditional prediction bounds, exact real models, finite arithmetic and physical observations answer different questions.