Separate the state, the rule, and what you observe.
By the end: Construct and iterate a deterministic state update.
This course assumes high-school science and basic algebra, not calculus or meteorology. Rates, slopes, and probabilities are explained as they appear. All numerical scenarios are constructed teaching examples, not current weather forecasts. Read in order for the first pass; later lessons use the earlier distinctions between state, rule, and uncertainty.
A state is the information a model needs now to calculate what happens next. Position alone is not enough to predict a moving ball: two balls at the same place can have different velocities. A weather state includes temperature, winds, moisture, and pressure across space, not just the temperature at one station.
An observation is a measurement of part of that state. A thermometer reading is not the full atmosphere. Even a deterministic rule cannot give a unique forecast from an incomplete observation unless you supply assumptions about the missing quantities.
A state belongs to a particular model, rather than to an object without qualification. Consider two connected tanks whose amounts are A and B liters. Suppose each update transfers one quarter of B into A, with no other inflow or outflow. The next amounts are A plus B divided by four, and three quarters of B. This rule needs both amounts. Knowing only the combined amount does not tell you how much transfers, because that transfer depends on the amount currently in B, not on the total.
Take two possible starting states with the same total of eight liters. In one, A is two and B is six. In the other, A is six and B is two. After one update, the first has three and a half liters in A, while the second has six and a half. Both still total eight. An instrument that reports only total water cannot distinguish these states, although their future individual tank amounts differ. This is missing information, not randomness in the transfer rule or a failure of arithmetic.
A and B are tank amounts in liters; primes mark the new amounts. Both calculations read old B.
| Case | Old A, B in liters | Next A, B in liters |
|---|---|---|
| First | 2, 6 | 3.5, 4.5 |
| Second | 6, 2 | 6.5, 1.5 |
| Total only | 8 | 8 |
Which quantity you want to predict changes what information is sufficient. For these sealed tanks, the total alone predicts the next total exactly, because transfer cannot change it. The total does not predict the next amount in A. A reduced state can therefore be adequate for one forecast and inadequate for another. In weather, predicting one station temperature and predicting a spatial wind field are different tasks. The tank example establishes this logical distinction without pretending that a two-number model represents the atmosphere or measures its uncertainty.
You can test a proposed state by asking whether two cases with the same proposed state can have different next values of the target quantity under the stated rule. Finding such a pair rejects sufficiency. Failure to find a pair in a few trials does not establish sufficiency for every possible case. For the tank total, algebra gives the stronger argument. Adding A plus B divided by four to three quarters of B recovers A plus B for every allowed starting amount, not merely the two examples.
A parameter has a different role from a changing state variable. In the tank model, the fraction one quarter remains fixed while A and B change. Replacing that fraction with one half changes the rule. Replacing the initial amount in B changes the starting state under the same rule. These are separate comparisons. If a valve setting itself changes according to the tank levels, the model needs a rule for that setting as well. It may then belong in an enlarged state rather than remain an externally fixed parameter.
| Calculation | Transfer to A, liters | Next total, liters |
|---|---|---|
| Both read old B | 6 / 4 = 1.5 | 3.5 + 4.5 = 8 |
| A reads new B | 4.5 / 4 = 1.125 | 3.125 + 4.5 = 7.625 |
| Difference | 0.375 too little | 0.375 missing |
The tank update also assumes that both new amounts use the same old B. If you first replace B by three quarters of B and then compute the transfer from that replacement, you transfer too little. Starting with B equal to six, that mistaken transfer is 1.125 liters instead of 1.5 liters. An implementation can obey its own deterministic instructions while implementing the wrong intended model. Before discussing predictability, establish which old quantities each calculation reads and whether the update conserves what the model says it should conserve.
We start with a dimensionless state x between zero and one. The logistic map calculates the next state by multiplying r, x, and one minus x. The parameter r controls the rule; it is held fixed during each run. The subscript n counts updates, not hours.
With r = 2 and x = 0.2, the first update gives 0.32, then 0.4352. Feed each result back as the next input. The diagram moves vertically to the curved update rule, then horizontally to the diagonal so that the output becomes the new input. The static table gives every illustrated step.
The logistic state is a number without a physical unit. A value of 0.4 does not mean forty percent humidity unless a separate model supplies that interpretation. The permitted interval simply restricts the possible model states. Write f for the operation that takes an input x and returns r times x times one minus x. Then f applied twice means applying f to its own output. It does not mean squaring the first output or multiplying that output by two. Repeated application produces a trajectory, the sequence of states indexed by update number.
For a new paper example, hold r at 1.5 and start at x equal to 0.4. First subtract 0.4 from one to get 0.6. Multiply 1.5 by 0.4 and then by 0.6 to get 0.36. The second input is now 0.36, so its complementary factor is 0.64. Their product with 1.5 gives 0.3456. The third update uses 0.3456 and 0.6544 and gives 0.33924096. Each subtraction must use the latest state, even though the multiplier r never changes during this trajectory.
xₙ is the dimensionless state after n updates; r = 1.5 is fixed. An update has no assigned physical duration.
State x, dimensionless
Update n, count
An update number records order, not a measured duration. The difference between n equal to zero and n equal to one is one application of the recurrence. A physical model might assign one update to one breeding season or one sampling interval, but this example assigns neither. Its first six updates are six algebraic operations in sequence. Taking longer to watch their drawing cannot change the results. Conversely, displaying all six at once does not make them simultaneous states of one run. Their indices preserve their sequence even in a static table.
A useful check is to compare a state with its immediate successor. At r equal to 1.5, subtracting x from the logistic output gives x times the quantity 0.5 minus 1.5 times x. For a positive state above one third, this difference is negative. For a positive state below one third, it is positive. This tells you the direction of the next change without computing an entire orbit. It does not yet say whether an update crosses one third, or establish convergence from every permitted starting value.
This particular run approaches one third from above. There is a direct check that it cannot cross below that value while its input lies between one third and 0.4. Subtract one third from the next state and factor the result as 1.5 times the quantity x minus one third times the quantity two thirds minus x. Both factors involving x are nonnegative on that interval. The next state therefore stays at least one third. Together with the negative change above one third, this explains the downward, bounded sequence in this example.
| Input x | Next x | Next minus current |
|---|---|---|
| 0.2 | 0.24 | +0.04 |
| 1 / 3 | 1 / 3 | 0 |
| 0.4 | 0.36 | −0.04 |
| 0.6 | 0.36 | −0.24 |
Rounding should happen after the next input has been calculated if the goal is to follow this recurrence as closely as the available arithmetic permits. If 0.3456 is replaced by 0.346 before the third update, the result becomes 0.339426 instead of 0.33924096. That difference is caused by changing the input through rounding. It is not a random disturbance to a physical population. A rule that deliberately rounds after every update is a different discrete model, which may be useful for studying numerical effects but should be identified as such.
x is dimensionless and n counts updates. r is fixed within a run. We restrict 0 ≤ r ≤ 4 and 0 ≤ x ≤ 1.
Each choice restarts from x = 0.2 and resets playback. Each pair of moves completes one update. Reset keeps r. The curve is the rule; the diagonal copies the output into the next input.
Completed updates: 0. Last completed state: 0.200000.
| n | State x |
|---|---|
| 0 | 0.200000 |
| 1 | 0.320000 |
| 2 | 0.435200 |
| 3 | 0.491602 |
| 4 | 0.499859 |
| 5 | 0.500000 |
| 6 | 0.500000 |
| 7 | 0.500000 |
| 8 | 0.500000 |
| Update | Input x | r × x × (1 − x) |
|---|---|---|
| 1 | 0.2 | 2 × 0.2 × 0.8 = 0.32 |
| 2 | 0.32 | 2 × 0.32 × 0.68 = 0.4352 |
The cobweb uses two axes for two roles played by the same kind of number. The horizontal coordinate is the input to one update. The vertical coordinate is that update's output. A point on the curved rule therefore represents a calculation, not two independent state variables. The diagonal marks pairs whose two coordinates agree. Moving horizontally to it copies the computed output onto the input axis for the next calculation. The horizontal move does not apply the logistic rule again. Only the next vertical move calculates another state from that copied input.
Choose the existing Alternating, r = 3.2 preset for a contrast with the settling example. The control restarts at 0.2, regardless of how far the previous drawing had progressed. The first vertical move reaches 0.512, because 3.2 times 0.2 times 0.8 equals 0.512. The following horizontal move reaches the diagonal at the pair 0.512, 0.512. At that point exactly one update is complete. The next vertical move computes 0.7995392. These coordinates describe the preset's algebra, not distances traveled by a physical particle.
| Move | Coordinate reached | Completed updates |
|---|---|---|
| Vertical to rule | 0.2, 0.512 | 0 |
| Horizontal to diagonal | 0.512, 0.512 | 1 |
| Vertical to rule | 0.512, 0.7995392 | 1 |
| Horizontal to diagonal | 0.7995392, 0.7995392 | 2 |
A time plot of that same run uses different coordinates. Its horizontal axis is the integer update number, and its vertical axis is the state at that update. The cobweb puts input and output next to each other to explain reuse of a result. The time plot puts successive results in order to expose alternation. Neither view adds information to the full ordered sequence, but each makes a different relation easier to inspect. Confusing these axes can make a horizontal cobweb move look like a period of constant physical state.
The player's two moves per update explain why its drawing position can advance while the completed-update counter stays unchanged. A partly drawn vertical segment is progress in presenting a calculation. It is not a fractional logistic update. Next step advances one drawing move, so two such moves complete one recurrence application. The speed control changes how fast those moves appear, not the multiplier r or the resulting numbers. The closed static trace lists all eight updates without requiring any inference from the speed or smoothness of the displayed motion.
Reset keeps the selected r and returns the drawing to its beginning. Selecting another preset changes r and restarts both the calculation and presentation from 0.2. These operations answer different questions. Reset lets you inspect the same sequence again. A preset comparison asks how another fixed rule evolves the same initial state. There is no control here for continuing an existing trajectory with a suddenly changed r. Such a switched rule is mathematically possible, but it is a separate experiment and would require specifying the update at which the switch occurs.
State x, dimensionless
Update n, count
The first few values at r equal to 3.2 suggest alternating high and low values, but eight updates do not establish an exact repeating cycle. The early values still change between successive visits to the high part of the graph. A rounded display can hide those changes. You can compare the first and third high values in the static trace to see this transient approach. The distinction matters because a repeated printed decimal is weaker evidence than an algebraic demonstration that an entire orbit returns to its starting state after a specified number of updates.
The same complete initial state and rule give the same future in this model. That says nothing yet about how errors in the initial state behave. Later lessons compare attraction and divergence. There is no random draw hidden in this experiment.
A state update is a modeling choice. This simple map can illustrate mechanisms, but it does not forecast population counts or weather merely because it has complicated outputs. Distinguish a mathematical consequence from evidence that the rule describes a particular real system.
Determinism describes the relation between complete inputs and outputs, rather than what an observer knows. Suppose the model uses r equal to two, but an observation places the starting state somewhere between 0.30 and 0.40. Each possible initial value has one specified next value. The observer nevertheless has several possible next values because the observation did not identify one input. Representing that uncertainty as an interval does not insert a random draw into the rule. It records which deterministic calculations remain compatible with the information supplied to the observer.
Over this particular interval, the logistic curve rises. The lower endpoint maps to 0.42 and the upper endpoint maps to 0.48, so all next states lie between those values. The initial interval has width 0.10; the next interval has width 0.06. Here the range contracts after one update. This calculation requires the curve to be monotone over the whole interval. If the initial interval crossed the curve's peak, checking only its two endpoints could miss the largest possible output. An interval forecast needs the rule between its endpoints as well.
Next x after one update
Possible initial x, dimensionless
For example, under r equal to two, the interval from 0.4 to 0.6 has equal endpoint outputs of 0.48. It would be wrong to conclude that all possible outputs equal 0.48. The included input 0.5 produces 0.5, which is the maximum. The correct next interval is therefore 0.48 to 0.5. The curve folds the two sides onto overlapping outputs. This loss of information about earlier states does not violate determinism, which requires a unique future from each state, not a unique past from each future state.
Uncertainty about r is a separate possibility. Fix the initial state at 0.4 but compare r equal to two with r equal to three. The next states are 0.48 and 0.72. In this comparison, the starting state is identical and the rules differ. A forecast discrepancy could therefore arise even when the initial state was known exactly. An analyst who changes both r and the initial state at once cannot attribute the changed result to either choice alone. Holding one fixed gives a controlled mathematical comparison, not evidence about actual atmospheric errors.
A limited forecast can also concern a property rather than one exact value. If the permitted starting interval is 0.30 to 0.40 and r equals two, the one-update statement that x exceeds 0.4 is true for every compatible state. The exact decimal value is uncertain, but that threshold statement is not. Another threshold, such as exceeding 0.45, cannot be decided uniformly from the same information. Forecast usefulness therefore depends on the requested quantity and tolerance, even before longer-term error growth enters the discussion. No probabilities follow from an interval alone.
| Question about next x | Answer for all compatible inputs |
|---|---|
| Above 0.4? | Yes, minimum is 0.42 |
| Above 0.45? | Mixed, outputs straddle 0.45 |
| Above 0.5? | No, maximum is 0.48 |
| Exactly 0.455? | Possible at initial 0.35, not required |
Finally, matching a few observed outcomes does not show that a recurrence is the right physical model. Different rules can agree at selected inputs and disagree elsewhere. Measurements also have units, spatial coverage, and error sources absent from this dimensionless map. To use a model for weather, one must connect modeled variables to observations and check forecasts on cases not used to choose the rule. The logistic examples establish how to reason from a stated recurrence. They do not establish that its parameters or update counts describe any particular weather process.
Two pendulums are at the same angle. Must they next move the same way?
Consider the direction of motion.
| Case | Angle | Angular velocity | Next motion |
|---|---|---|---|
| A | 10° | Negative | Toward smaller angles |
| B | 10° | Positive | Toward larger angles |
No. One can move left and the other right. Angle and angular velocity are needed for a basic pendulum state. Equal observed angle does not imply equal complete state.
For r = 2 and x = 0.25, calculate the next two states.
The second update starts from your first result.
| n | x |
|---|---|
| 0 | 0.25 |
| 1 | 0.375 |
| 2 | 0.46875 |
The first state is 2 × 0.25 × 0.75 = 0.375. The second is 2 × 0.375 × 0.625 = 0.46875. Reusing 0.25 would calculate the first update twice rather than evolve the system.
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.
Secondary overview of deterministic dynamics and sensitive dependence. Not a proof that every irregular numerical trace is chaotic.
Recurrence, fixed points, period doubling, and periodic windows. Selected explanatory passages reviewed; the plots here are generated independently.