Change a rule parameter and inspect possible long-run behavior.
By the end: Distinguish parameter changes from time evolution.
Each vertical slice is a different experiment
A bifurcation is a qualitative change in the dynamics as a parameter changes. Here the horizontal axis is r, not time. For each r, start at 0.213, perform 900 updates, and plot the last 48 values. A vertical slice summarizes late states for that one parameter.
One narrow branch suggests a fixed point; two suggest a two-cycle. As r increases past three, the attracting fixed point gives way to a two-cycle. Further period doublings create four, eight, and more branches, with increasingly narrow parameter intervals.
Separate an orbit from a parameter scan
A time trace and a bifurcation diagram answer different questions even when they use the same recurrence. A time trace holds r fixed and places update number on the horizontal axis. A bifurcation diagram holds neither update number nor one continuing orbit on that axis. It collects late states from separate calculations. To reconstruct one dot, choose its r, start at the stated initial value, apply the recurrence repeatedly, and retain a selected late state. The dot's vertical coordinate is a state, not an uncertainty interval around a forecast.
Consider two additional runs starting at 0.213, one with r = 2.9 and another with r = 3.1. Their first outputs are 0.4861299 and 0.5196561. These follow by multiplying the same input product, 0.213 times 0.787, by different parameters. The outputs become inputs for their respective second updates. Neither calculation borrows a state from the other. Keeping these runs separate makes the construction explicit before compressing hundreds of updates into a vertical slice. All values here are dimensionless, and an update has no assigned duration in seconds.
Two independent short orbits
State x, dimensionless
Update number, dimensionless
Solid: r = 2.9
Dashed: r = 3.1
Both runs start at 0.213 and retain updates zero through five. Connected points identify successive discrete model states, not continuous motion between them. This short horizon shows different rules acting on one start, not established late behavior.
At r = 2.9 the late values approach about 0.6551724. At r = 3.1 they alternate near 0.5580141 and 0.7645665. The first five updates do not establish either result. They show transient behavior, meaning behavior before the late pattern becomes representative of this start. A plot retaining those first outputs would combine the approach with the eventual pattern. That combination can be useful for a time trace, but it answers a different question from the late-state diagram. Discarding early values is therefore a choice about what to display, not an alteration of the recurrence.
The implemented scan performs 900 updates for each r and retains states numbered 853 through 900. There are 48 retained values because both endpoints count. It does not discard 900 updates and then perform another 48. This distinction changes the computational recipe and matters when comparing two implementations. A long calculation can still be too short near a slowly attracting regime. The chosen cutoff defines a reproducible finite experiment; it does not guarantee that every sampled orbit has reached its limiting behavior or that unrepresented starts behave identically.
Each vertical slice contains 48 plotted states, regardless of how many different values the eye distinguishes. If a run alternates between two well-separated levels, approximately 24 marks can overlap at each level. A single visible point may conceal all 48 marks at an attracting fixed point. A broad slice may instead contain many distinct retained values. Thus point count, distinct state count, and visible darkness are different quantities. The plotting recipe fixes the first, while dynamics and display resolution influence the others. None alone specifies the probability of a future physical event.
What 48 marks can represent
Exact late pattern
Marks per level
Distinct levels
Fixed point
48
1
Two-cycle
24
2
Four-cycle
12
4
Eight-cycle
6
8
Every slice has the same retained denominator of 48 states. These are idealized allocations for exact repeating patterns whose periods divide 48. Overlap can hide repeated marks; their darkness is not a calibrated frequency estimate for weather.
An orbit can visit the same plotted height at different update numbers, and the bifurcation diagram removes that ordering. Two vertically separated marks do not reveal which comes first. To recover order, return to the earlier nonlinear-feedback experiment and inspect its time trace at the same fixed parameter. The parameter slider selects a new independent run rather than advancing that trace. In this interface it recomputes the selected values from 0.213, while the background scan remains unchanged. There is no playback clock in this experiment, so moving the slider faster has no meaning for the model's dynamics.
Do not assume increasing disorder everywhere
The first period-doubling cascade accumulates near r = 3.56995. Beyond it the diagram contains irregular bands and periodic windows. Around r = 3.83 a period-three window is visible. Increasing r does not monotonically increase a single measure of disorder.
The plot has finite parameter spacing and a finite burn-in. It can miss narrow windows or retain slow transients. A dense band is not an exact list of all possible states and its darkness is not a calibrated probability density.
Resolve branches without mistaking them for noise
A branch is a collection of late states indexed by parameter, not a path followed while one orbit changes its rule. On a stable two-cycle there are two such collections. Increasing r slightly shifts both cycle levels. Algebra can identify them without reading pixel positions. For r above three, form the square root of the product of r minus three and r plus one. Subtract that root from r plus one and divide by twice r for the lower level. Addition instead of subtraction gives the upper level. Factoring the two-update return equation and removing fixed-point factors gives these two roots.
Form the cycle discriminant
For r above three, D is positive. The lower and upper states use the same positive square root.
Separate the two branch formulas
a and b are dimensionless states of the same two-cycle, not times.
For example, r = 3.3 gives a product of 0.3 times 4.3, or 1.29. Its square root is about 1.1357817. Dividing 4.3 minus this root by 6.6 gives 0.4794270; using the plus sign gives 0.8236033. Substitution into the logistic rule sends each level to the other, apart from displayed rounding. Neither level is individually a fixed point. The pair is invariant as a set because one complete update exchanges its members. Two updates return each member to itself, which explains the term two-cycle.
Calculated two-cycle levels
Cycle level x, dimensionless
Parameter r, dimensionless
Solid: Lower level
Dashed: Upper level
These algebraically calculated levels use three fixed parameters in the attracting two-cycle regime. Lines interpolate between sampled parameter values; they are neither time trajectories nor exact branch shapes between samples. Each level maps to its partner under its own r.
Finding a cycle algebraically does not establish that nearby starts approach it. That requires a stability calculation. A small displacement near a state x is approximately multiplied by r times the quantity one minus twice x during one update. For a cycle, multiply these factors along the complete return. The two-cycle return factor simplifies to negative r squared plus twice r plus four. At r = 3.3 this is negative 0.29, whose magnitude is less than one. Small errors measured every two updates therefore contract locally, with alternating signs between returns.
Cycle discriminant at r = 3.3
D is the quantity inside the two-cycle square root; a and b are lower and upper cycle states, all dimensionless.
Recover the two branches
The minus choice gives a ≈ 0.4794270; the plus choice gives b ≈ 0.8236033.
State the general return factor
This is the first-order factor over two updates for the algebraic two-cycle. Existence of that cycle alone does not imply attraction.
Check local return stability
M multiplies a sufficiently small signed error over two updates. Its magnitude below one establishes local, not global, attraction.
At r = 3.5 the same return factor is negative 1.25. The algebraic two-cycle still exists, but it no longer attracts nearby states through small perturbations. A late-state scan can therefore stop displaying a branch even though the corresponding exact orbit has not ceased to be a solution. The scan shows behavior reached from its chosen start after its finite settling interval. It is not a catalogue of every unstable orbit. This distinction explains why disappearance from a numerical diagram is weaker evidence than nonexistence established through an equation.
Parameter spacing introduces a separate limit from settling time. The background samples r = 2.8 plus an integer multiple of 0.005, ending at four. Its 241 samples include both endpoints. A feature narrower than that spacing might receive no sampled parameter at all. Increasing the number of retained states at existing parameters would not repair this omission. Conversely, adding parameter samples would not remove an insufficiently decayed transient. The two numerical choices address different sources of ambiguity, even when both produce a diagram that looks more detailed to the eye.
Local return factors for the two-cycle
r
Two-update factor
Local conclusion
3.1
0.59
Attracting
3.3
−0.29
Attracting
3.4
−0.76
Attracting
3.5
−1.25
Repelling
The factor applies to a sufficiently small displacement after two updates, not after one. Its sign indicates reversal between returns. Values below unit magnitude support local attraction; they do not assert convergence from every allowed starting state.
Near a transition, a return factor with magnitude close to one can make the approach unusually slow. Repeated multiplication by 0.99 leaves about 0.366 of an initial small error after 100 returns. Multiplication by 0.5 leaves less than one thousandth after ten returns. These are illustrative local contractions, not estimates of the implemented scan's error at a particular r. They explain why one fixed settling budget can be generous in one parameter interval and inadequate in another. A finite scan needs interpretation even when every arithmetic operation follows the intended recurrence.
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.
Selected r = 3.2; last six values: 0.51304, 0.79946, 0.51304, 0.79946, 0.51304, 0.79946. Changing the slice does not change the blue sample. It recomputes the selected run from the same start.
241 parameter samples, 900 updates each, last 48 plotted. Neither dense shading nor finite repeats prove a continuous distribution or exact cycle.
r
Typical late behavior from the chosen start
2.8
One attracting value
3.2
Two alternating values
3.5
Four values
3.83
Period-three window
Parameter regimes are not time stages of one run.
Audit the finite sampling experiment
The selected slider and the blue background have different numerical resolutions. The slider advances by 0.01, while the background advances by 0.005. A blue column can therefore lie between two selectable slider positions. The named presets choose particular parameter values, and the selected run starts again at 0.213 after either a preset or slider change. It does not reuse the final state of the preceding selection. This behavior permits controlled comparisons because the initial condition stays fixed. It does not test dependence on initial conditions, since this experiment provides no initial-state control.
The background computes 241 times 900 updates, or 216900 recurrence evaluations, and draws 241 times 48 retained marks, or 11568 marks. The selected slice separately computes another 900 updates. These counts describe the teaching implementation, not a benchmark of atmospheric prediction or browser speed. Repeated values remain repeated marks in the data. A renderer can place several of them on the same pixel. Counting visible dots on a screenshot will therefore underestimate the number of retained states, sometimes by a large factor at simple attracting patterns.
Count parameter samples
N counts fixed-parameter runs; both endpoints are included. These are operation and mark counts, not accuracy or runtime measurements.
Count recurrence evaluations
Count retained marks
Each run retains updates 853 through 900, inclusive: 900 − 853 + 1 = 48. Repeated states still count as marks.
Counts in the displayed experiment
Quantity
Calculation
Count
Parameter samples
240 intervals + 1
241
Background updates
241 × 900
216900
Background marks
241 × 48
11568
Selected updates
1 × 900
900
Counts refer to one construction of the fixed background, plus one selected slice. Retained states include repetitions. They measure the sampling budget, not numerical accuracy, execution time, or the number of distinct attracting states.
A useful comparison within the available controls is r = 3.4 against r = 3.55. The slider can select both, although neither needs a dedicated preset. The first gives two late levels near 0.451963 and 0.842154 from the fixed start. The second gives eight distinct rounded late levels in this finite calculation. This comparison concerns the computed samples, not a claim that all larger parameters have more levels. It connects a visible branch count to an explicit retained-value check while avoiding the unsupported assumption that increasing r always increases irregularity.
Period counting requires a definition of equality. Exact arithmetic would ask whether the state returns exactly after a stated number of updates. Floating-point arithmetic stores a finite representation, and a printed decimal discards still more information. Two values displayed as 0.842154 may differ beyond the sixth decimal. Conversely, tiny numerical differences can obscure an attracting pattern that is clear at the diagram's scale. For this reason, describe rounded repetitions with their precision and horizon. An exact period claim needs algebra or stronger analysis than matching a few printed digits.
The r = 3.55 example illustrates how selecting only the last six values can conceal a longer repeating pattern. Those six are approximately 0.540475, 0.881684, 0.370326, 0.827805, 0.506031, and 0.887371. The retained 48 states contain eight rounded levels, but six displayed entries cannot show a complete eight-update return. This is a limitation of the selected textual readout, not a contradiction with the background plot. The readout provides concrete samples. It is not designed to identify every possible period automatically or to certify an invariant set.
Same run, different retained summaries
Parameter
Last six: distinct rounded levels
Last 48: distinct rounded levels
2.9
1
1
3.4
2
2
3.55
6
8
3.83
3
3
These independently computed summaries start at 0.213 and end at update 900. Distinct counts use rounding to six decimal places. They describe finite samples, not proofs that the mathematical orbit has exactly the reported period.
The background also removes the identity of each retained update once its dot is placed. An eight-cycle and an irregular orbit visiting eight narrow neighborhoods might look similar at low resolution. A longer ordered trace would address their return sequence, while an algebraic calculation could address exact recurrence. Those are additional investigations rather than functions of the existing slider. Keeping that boundary explicit prevents a useful interactive illustration from becoming an imagined analysis tool. The current diagram supports bounded observations about its stated recipe, followed by questions that require evidence beyond the picture.
Use the diagram to ask better questions
Select a vertical slice and inspect its six printed late values. For a longer ordered trace at the same r, return to the earlier nonlinear-feedback experiment. Ask whether that trace alternates or stays near a few values. Testing another initial state requires a separate calculation; this widget fixes the start at 0.213.
A parameter shift changes the rule. An initial-state perturbation changes where a run starts under the same rule. Both matter in forecasting, but they are different experiments and need different interpretations.
Specify what a diagram claim would require
A bifurcation concerns a change in the qualitative behavior of a rule as its parameter changes. A large vertical shift between two samples need not be a bifurcation. An attracting fixed point can move continuously while keeping the same local stability type. For example, the nonzero fixed-point formula gives 0.6428571 at r = 2.8 and 0.6551724 at r = 2.9. Their locations differ, but both are locally attracting. The meaningful change is not merely that a plotted number moved; it is that the kind or stability of invariant behavior changed.
One way to distinguish location from stability is to calculate both separately. The fixed point is one minus the reciprocal of r. Its local multiplier is two minus r. At r = 2.9 the multiplier is negative 0.9, while at r = 3.1 it is negative 1.1. Small errors contract in the first case and expand in the second, at least while the local approximation applies. Both fixed points still solve the same fixed-point equation. Their changed stability helps explain why the finite scan follows a different late pattern on opposite sides of three.
A fixed point can persist while losing attraction
r
Fixed point x
Local multiplier
2.8
0.6428571
−0.8
2.9
0.6551724
−0.9
3.0
0.6666667
−1.0
3.1
0.6774194
−1.1
The nonzero fixed-point locations follow one minus one divided by r. Multipliers describe sufficiently small errors over one update. The r = 3 row is a linear boundary case; unit magnitude alone does not settle nonlinear behavior there.
An exactly initialized unstable fixed point remains fixed in exact arithmetic. That statement does not conflict with nearby trajectories leaving its neighborhood. Stability asks what happens to a displacement, not what happens when the displacement is exactly zero. In numerical arithmetic, representing the fixed point only approximately may introduce such a displacement. The scan's start of 0.213 is not the unstable nonzero fixed point for these examples. Consequently, failing to see that fixed point in its late-state output is expected and says nothing about whether it remains an exact mathematical solution.
Another distinction is between varying a parameter across independent runs and driving a system with a changing parameter. Suppose a separate model specified r at every update rather than keeping it constant. Then the current state would depend on the whole earlier parameter sequence. It could lag behind the fixed-parameter attracting levels or carry a transient into a new regime. The present bifurcation diagram supplies no such sequence and computes no such lag. A static collection of late states cannot by itself predict the response to a rapid or gradual external change.
A periodic window also needs careful interpretation. The three late levels at r = 3.83 are about 0.1561493, 0.5046665, and 0.9574166 in cyclic order for this computation. They are a specific numerical example of order within a region often described broadly as chaotic. Their occurrence does not imply that nearby parameters share exactly those values or even the same attracting pattern. Nor does it assign a chance that an actual population or atmosphere will follow three phases. The recurrence, parameter, initial condition, and sampling procedure define the scope of the statement.
A numerical return inside the three-cycle window
Input x
Next output, approximately
0.1561493
0.5046665
0.5046665
0.9574166
0.9574166
0.1561493
At fixed r = 3.83, these rounded late states come from the run starting at 0.213. Each output supplies the next input. The three substitutions illustrate a return; finite rounded agreement is not a proof of exact periodicity.
The strongest justified reading of the diagram combines its construction with separate mathematical results. The picture locates candidate regimes and makes comparisons visible. Substitution verifies a proposed invariant orbit. Local return factors assess small disturbances near that orbit. None of these steps automatically proves global attraction or classifies every remaining parameter. This division of evidence is useful when reading an expert's claim about a transition. Ask whether the claim concerns one computed trace, local behavior around a known solution, or a theorem about a whole parameter interval. Those statements require different support.
Exercises
Interpret the horizontal axis
A reader says the system evolves from r = 2.8 to 4 during the plot. What is wrong?
Hint
Each slice restarts the calculation.
Worked answer
Experiment
What changes between updates?
One orbit
State x; r stays fixed
Bifurcation sweep
Separate orbits at separate r values
Two distinct constructions.
The horizontal axis labels different fixed-parameter experiments. No run changes r over time. To model an externally changing r, we would need an explicit rule for that change and a different simulation.
Inspect a missing window
A coarse plot shows no period-three window. Can it disprove that one exists?
Hint
Consider the spacing between sampled parameter values.
Worked answer
Sampled r
Result
3.80
Outside the example window
3.90
Outside the example window
3.83, not sampled above
Can show period-three behavior
Sampling can miss an interval.
No. A narrow window can fall between samples. Increase parameter resolution near a candidate interval, allow enough transient removal, and compare with mathematical analysis. More dots alone are not a proof.
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.