Follow two nearly equal states under exactly the same rule.
By the end: Explain a conditional predictability horizon without treating chaos as randomness.
Compare trajectories, not just one irregular line
Both trajectories use the same r and the same arithmetic. Only the initial state differs. At r = 3.9, a tiny initial separation can become a large difference after enough updates. At r = 2.8, nearby interior starts instead converge toward the same attracting point.
Chaos concerns deterministic dynamics with sustained sensitive dependence and nonperiodic behavior on a relevant invariant set. Definitions differ in mathematical detail. The bounded logistic example helps distinguish this from unlimited exponential growth away from an unstable point. Stretching can separate nearby states while folding keeps later states within a bounded interval. Folding also means some distinct starts can merge, so sensitivity does not mean every pair separates monotonically.
Derive what one update does to a disturbance
A pairwise comparison needs a reference state, a disturbed state, and a rule shared by both. Write the disturbed state as x plus d, where d is a signed displacement. Subtract the reference output from the disturbed output after applying the logistic rule to each. Expanding the products gives a new displacement equal to r times d times the quantity one minus twice x minus d. This identity is exact for the recurrence, provided both inputs lie in the allowed interval. It shows why separation depends on location as well as its starting size.
Take the additional example x = 0.31, d = 0.0001, and r = 3.9. The factor one minus twice x minus d is 0.3799. Multiplying by 3.9 and by the displacement gives 0.000148161. Direct calculation gives reference output 0.83421 and disturbed output 0.834358161, confirming that subtraction. The initial absolute separation has increased by a factor of 1.48161 in this update. No random kick was added. Both outputs follow the same deterministic multiplication rule; their different inputs suffice to create different later states in this example.
Exact displacement rule
x is the reference input, d the signed input gap, and d′ the signed output gap under the same r. All are dimensionless.
Evaluate the location factor
Calculate the new gap
An exact one-update displacement calculation
Quantity
Calculation
Value
Reference output
3.9 × 0.31 × 0.69
0.83421
Disturbed output
3.9 × 0.3101 × 0.6899
0.834358161
Signed difference
Second output minus first
0.000148161
Growth factor
Difference divided by 0.0001
1.48161
The two dimensionless starts differ by 0.0001 under the same r = 3.9 rule. Subtraction and the expanded displacement identity agree. These calculations describe one model update, with no physical time interval or measured uncertainty attached.
The next update acts at different input locations. The reference is now above one half, so the quantity one minus twice x is negative. A positive signed displacement can become negative after updating, meaning that the disturbed trajectory has moved below the reference. Absolute difference removes that sign. After two updates the example's signed displacement is approximately negative 0.0003863173, while its absolute separation is positive 0.0003863173. A separation graph therefore cannot tell which trajectory is higher. Reading it together with the two state traces restores that information without changing either calculation.
The same pair has absolute separation about 0.0001180938 after three updates. It shrinks relative to the preceding update even though this is the sensitive r = 3.9 regime. Location near the top of the logistic curve produces weak output differences because that curve is nearly flat there. A long-run tendency to amplify small disturbances need not hold at every individual update. The plotted error can rise and fall, and those declines are part of the deterministic dynamics. They do not require a resetting event, measurement correction, or hidden random input.
There is an even stronger counterexample to monotonic separation. The inputs 0.27 and 0.73 have the same product x times one minus x. At r = 3.9 both map to 0.76869. Once they agree, later exact updates also agree because each subsequent input is identical. This happens because the map folds the interval, sending two distinct inputs to one output. It prevents a claim that every distinct pair must keep separating. Sensitive dependence describes behavior of nearby possibilities on a relevant set, not an absolute promise about every possible pair of starts.
Separation can shrink during a sensitive run
Absolute state separation, dimensionless
Update number, dimensionless
Solid: Absolute separation
Starts are 0.31 and 0.3101, with r fixed at 3.9. Connected points show absolute separation at consecutive discrete updates through update four. Straight segments guide the eye; the map defines no intervening continuous error trajectory or atmospheric timescale.
When d is very small, the term containing d squared contributes little compared with the term proportional to d, unless the latter is itself near zero. Dropping that squared term gives the local multiplier r times one minus twice x. This is an approximation for small disturbances, not a replacement recurrence valid at every separation. Near x = 0.5 the linear multiplier vanishes and the squared term matters. Farther apart, the exact expression should replace the approximation. High-school expansion of two products thus explains both the useful linear rule and one of its limits.
Separate the linear and quadratic changes
The signed gap d is small only relative to the local terms being compared. Near x = 1/2 the linear term vanishes.
Keep only the first-order term
This approximation omits the quadratic term; it does not assert a constant long-run growth rate.
Growth rates and useful horizons
Over a limited regime, a small error can grow approximately exponentially. If it doubles every two model-time units, reducing initial error by a factor of eight buys roughly three extra doublings, or six time units. It does not buy an arbitrarily long exact forecast. In the table, log₂ asks how many doublings produce the stated ratio, so fractional answers describe part of a doubling interval.
The Lyapunov exponent describes a long-run average exponential rate for infinitesimal disturbances along a trajectory. Local growth can fluctuate. Our plot shows a finite pairwise separation, not a validated Lyapunov-exponent estimator. Saturation matters because both logistic states stay between zero and one.
Calculate a horizon from a stated error model
A predictability horizon needs a definition of acceptable error before it needs a growth formula. Suppose an invented error magnitude doubles every 1.5 dimensionless model-time units, begins at 0.002, and becomes unacceptable above 0.064. The tolerance divided by the initial error is 32. Since five doublings multiply an error by 32, the error reaches tolerance after 7.5 model-time units. This example stipulates its growth rate rather than estimating it from the logistic experiment. Its time coordinate is a separate teaching assumption, not a conversion of updates into weather forecast days.
Now reduce the stipulated initial error to 0.0005 while keeping the same tolerance and growth law. The required ratio becomes 128, which is seven doublings. Reaching tolerance takes 10.5 model-time units. A fourfold improvement in initial precision has bought two more doublings, or three additional units, rather than multiplying the horizon by four. This follows because repeated multiplication undoes a fixed proportional improvement after a fixed number of doubling intervals. The arithmetic explains why large investments in initial precision can produce useful but limited extensions under an exponential approximation.
Invert the assumed growth law
H is time to tolerance ε, e₀ is initial error, and τ = 1.5 dimensionless model-time units is the stipulated doubling time. This is not a measured logistic growth rate.
Original initial error
Fourfold smaller initial error
Additional lead time
Two stipulated exponential error envelopes
Assumed error magnitude, dimensionless
Stipulated model time, dimensionless
Solid: Initial error 0.002
Dashed: Initial error 0.0005
Errors double every 1.5 dimensionless model-time units; the common tolerance is 0.064. Lines interpolate these exponential samples and are not exact curves between points. Values after tolerance crossing only illustrate the assumed law, not validated logistic or atmospheric growth.
The general calculation divides tolerance by initial error, takes the base-two logarithm of that ratio, and multiplies by doubling time. A logarithm answers how many repeated doublings produce a given factor. It can return a fraction when the ratio is not a whole power of two. For initial error 0.003 and tolerance 0.05, the ratio is about 16.6667. Its base-two logarithm is about 4.0589, giving 6.0883 units at the stipulated doubling time. This fractional horizon assumes a continuous exponential envelope, not an additional fractional logistic-map update.
Changing the tolerance also changes the horizon without changing the underlying dynamics. With initial error 0.002, a tolerance of 0.016 allows three doublings and a tolerance of 0.064 allows five. The difference is three model-time units under the assumed rate. A forecast user who needs a precise location can therefore have a shorter useful horizon than one who needs only a broad regional quantity. The toy calculation does not quantify that atmospheric difference. It establishes why any claimed horizon must specify both the predicted quantity and its acceptable error.
The exponential envelope applies only while small-error growth is a reasonable approximation. The logistic state interval gives an absolute separation bound of one, so an envelope that grows without limit must eventually fail there. Folding can also reduce an actual pair's separation long before that bound. A horizon derived from the envelope is consequently conditional, not a guarantee that every pair crosses tolerance at the same update. Treating a first crossing in one finite pair as a universal predictability limit would confuse a sample path with a model of uncertainty growth.
Tolerance changes useful lead time
Tolerance
Required doublings
Time to tolerance
0.008
2
3 units
0.016
3
4.5 units
0.032
4
6 units
0.064
5
7.5 units
All rows assume initial error 0.002 and doubling time 1.5 dimensionless model-time units. The tolerance-to-initial-error ratio determines the required doublings. These are conditional calculations, not weather measurements or forecasts of when a particular pair will first disagree.
A growth rate can vary with time and direction in a system with several variables. Even in the scalar map, the local multiplier changes as x changes. Multiplying local factors over a short interval gives a finite-interval estimate, while averaging their logarithmic magnitudes over long trajectories relates to a Lyapunov exponent. The logarithm appears because it converts repeated multiplication into addition. The existing separation plot performs neither estimation. Its absolute differences are finite pairwise results, so a visible rise alone does not justify attaching a measured exponent or a universal doubling time to the experiment.
Inspect the rules and their result
The logistic update
x is dimensionless and n counts updates. r is fixed within a run. We restrict 0 ≤ r ≤ 4 and 0 ≤ x ≤ 1.
Starts: 0.213 and 0.213001000. Initial separation: 10 to the power −6. Changes restart both sixty-update calculations; neither trace is continued.
Solid blue: Reference. Dashed amber: Perturbed. Solid blue: Absolute separation.
Difference after 60 updates: 7.582e-1. A line near zero can be below display resolution, not mathematically zero.
Initial error
Required doublings
Approximate horizon
0.01
log₂ 10 ≈ 3.32
6.64 units
0.00125
log₂ 80 ≈ 6.32
12.64 units
Assumed doubling time two units, tolerance 0.1.
Read the paired experiment at its actual resolution
The sensitivity controls offer two fixed rule parameters, 2.8 and 3.9. The initial-separation presets and slider choose an exponent between three and nine. Both trajectories then begin again, with the reference at 0.213 and the comparison at 0.213 plus ten raised to the negative chosen exponent. The exponent describes the gap, not the state itself. Increasing it by one divides the starting gap by ten. Each change recomputes sixty discrete updates. No trajectory carries its ending state into the next selection, and no animation speed sets the model's update interval.
The two state traces and the absolute-separation trace share a fixed vertical range from zero to one. This allows comparable magnitudes between selections, but it hides very small differences at ordinary screen resolution. An error of one millionth occupies one millionth of the plot height. A line drawn against zero therefore need not indicate exact equality. The same issue affects two state traces lying on top of each other. Their visible overlap is a statement about the drawing scale until a numerical difference establishes more about the underlying stored states.
What the exponent changes
Exponent
Initial gap
Comparison start
3
0.001
0.214
6
0.000001
0.213001
9
0.000000001
0.213000001
The reference is always 0.213, and the comparison adds the displayed gap. These are three available preset selections before any updates. Each row defines a separate deterministic sixty-update experiment rather than progressively refining a run already in progress.
A useful controlled comparison keeps the initial-gap exponent fixed and changes only the dynamics preset. In the attracting case, an early increase need not prevent later contraction. For the separate starts 0.31 and 0.3101 under r = 2.8, the gap first grows from 0.0001 to 0.000106372, then falls to about 0.0000589567. Thus one growing step does not establish sustained sensitive dependence. The conclusion depends on evolution along the trajectory rather than a single local multiplication. These additional starts are paper calculations; the interface itself fixes its reference start at 0.213.
The displayed difference after update sixty is an endpoint statistic. It cannot reveal the largest earlier separation or the first update at which a chosen threshold was exceeded. A pair might separate substantially and later approach again. Reading the full error trace adds information about that history, but its fixed scale still limits small-gap comparisons. A threshold-crossing diagnostic would require a threshold definition and a calculation over all stored errors. Those are reasonable additional analyses, but they are not controls supplied by this experiment. The existing endpoint should not be silently relabeled as a forecast horizon.
Initial uncertainty and numerical rounding are different sources of disagreement. Here the chosen initial difference is deliberate, while both runs use the same floating-point arithmetic. Rounding can affect later results, especially after sensitivity amplifies small computational differences. That does not turn rounding into physical weather noise. Nor does identical machine output prove that physical initial uncertainty is absent. The pair experiment isolates one stipulated perturbation within one arithmetic implementation. It supplies evidence about that implemented recurrence, while a claim about atmospheric uncertainty needs observation errors, model limitations, and the relevant forecast quantity.
One early increase does not classify the dynamics
Update
Gap at r = 2.8
Gap at r = 3.9
0
0.000100000
0.000100000
1
0.000106372
0.000148161
2
0.000058957
0.000386317
3
0.000056976
0.000118094
4
0.000037207
0.000432020
These independent calculations use starts 0.31 and 0.3101, unlike the interface's fixed reference. Absolute gaps at selected updates show that both parameters can amplify the first gap. Four updates alone do not establish chaos or long-run convergence.
Holding r fixed in each comparison is essential for identifying the question being tested. If one trajectory used a different r, later disagreement would mix a changed initial state with a changed evolution rule. An ensemble forecast may deliberately represent both kinds of uncertainty, but this widget does not. Its simple design lets the learner attribute the imposed difference to a starting-state perturbation before considering more realistic models. The interpretation remains conditional on the shared rule. It is not a demonstration that initial uncertainty is the only important limitation on real weather prediction.
Avoid the butterfly misconception
Sensitive dependence does not mean every tiny cause necessarily produces a particular disaster, nor that one can choose the distant outcome by a tiny intervention. It describes limits on tracking trajectories when initial information is imperfect.
A forecast horizon depends on the variable, acceptable error, initial uncertainty, model error, and situation. The iteration count in this map is not a number of days of useful weather prediction. There is no universal horizon established by this toy model.
Distinguish uncertain trajectories from controllable outcomes
A small disturbance can be amplified without specifying the sign or shape of a distant consequence. The logistic difference identity already shows that signs can reverse as the orbit moves across the curve. In a many-variable system, disturbances can also change direction while their size grows. Knowing that an error may become large is therefore weaker than knowing which later event will occur. A claim that a particular tiny intervention would prevent a particular distant storm would require a causal model and reliable outcome comparison, not merely evidence of sensitive dependence.
Uncertainty describes a set of plausible initial states rather than just the distance between one chosen pair. Suppose a scalar initial state is known only to lie between 0.2 and 0.4, with r = 3.9. On this interval the logistic rule is increasing, so its smallest and largest outputs occur at the endpoints. They are 0.624 and 0.936. The output interval has width 0.312, compared with an input width of 0.2. This exact one-update bound requires no random sampling and assigns no probabilities to different positions within either interval.
Propagating an interval below the peak
Input x
Output 3.9x times one minus x
0.2
0.624
0.3
0.819
0.4
0.936
For r = 3.9, the rule is increasing throughout the initial interval from 0.2 to 0.4. These one-update values locate its output bounds exactly. No probability density, physical unit, or later-time error growth is supplied by this interval calculation.
Endpoint propagation fails when the interval crosses the peak at x = 0.5. For initial interval 0.4 through 0.6 under r = 3.9, both endpoint outputs are 0.936, but the middle input produces 0.975. The full output interval therefore extends from 0.936 to 0.975 rather than collapsing to a single value. Following only two endpoints would miss possible interior outcomes. This is another consequence of folding. A pairwise trace is useful for illustrating separation, but it is not automatically a valid description of how a whole uncertainty region evolves.
Interval width also does not determine probability. The inputs might concentrate near the center, cluster near one endpoint, or follow another distribution inside the same stated bounds. Those alternatives share an uncertainty interval yet can produce different output frequencies. An ensemble tries to represent selected plausible possibilities, with assumptions about how they are generated and weighted. The two-trajectory experiment supplies neither a distribution over starts nor weights. It cannot turn a fraction of its two lines above a threshold into a calibrated probability for the real atmosphere or even for an unspecified population of logistic starts.
Improved initial information still matters despite finite horizons. It can narrow the region of plausible near-term outcomes or delay threshold crossing under a useful growth approximation. Model improvement addresses a different error source by changing how states evolve. These benefits need not extend every forecast quantity equally. A regional average and an exact local extreme can respond differently to the same uncertainty. The map has only one state variable and no geography, so it cannot estimate those differences. It teaches the reasoning that a horizon belongs to a specified question, error standard, and model.
Equal endpoint outputs can hide interior possibilities
Input location
Input x
Output
Left endpoint
0.4
0.936
Interior
0.45
0.96525
Peak
0.5
0.975
Interior
0.55
0.96525
Right endpoint
0.6
0.936
All rows use r = 3.9 for one update. The interval from 0.4 to 0.6 crosses the rule's maximum. Its output bounds are 0.936 and 0.975, despite equal endpoint outputs; no probability weights follow from these bounds.
Determinism means that a fully specified state and rule select a unique next state. Practical prediction adds requirements about how accurately that state is known and how well the rule represents the system of interest. Those requirements can fail even when the mathematical update remains unique. Conversely, a sensitive model need not make every short forecast useless or every aggregate statistic unknowable. The evidence developed here concerns bounded comparisons and conditional error growth. Extending it to a forecast service requires independent checks of uncertainty representation and performance on the quantities people actually use.
Exercises
Buy additional lead time
Errors double every three hours in a stipulated early-growth regime. Initial uncertainty falls from 0.08 to 0.01. How much extra lead time to the same larger tolerance?
Hint
The improvement is a factor of eight.
Worked answer
Elapsed time
Reduced initial error after growth
0 hours
0.01
3 hours
0.02
6 hours
0.04
9 hours
0.08
Three doublings erase the eightfold improvement.
Eight is two cubed. It buys three additional doublings, or nine hours, while the exponential approximation holds. This is conditional arithmetic, not an estimated atmospheric doubling time.
Explain bounded disagreement
Two logistic trajectories disagree by 0.8. Should their separation keep doubling forever?
Hint
Both states stay in [0,1].
Worked answer
Quantity
Bound
First state
0 to 1
Second state
0 to 1
Absolute difference
0 to 1
The state space constrains the error.
No. Their absolute separation cannot exceed one. Exponential growth is an early small-error approximation, not an unlimited law. Later separations can shrink and grow again.
Sources and further reading
Selected reference passages support the definitions and modeling distinctions below. The examples, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.