Compare a state that stays put with nearby states that move.
By the end: Explain local attraction using the error multiplier.
A fixed point has next state equal to current state. For the logistic map, zero is always fixed. For r greater than one, another fixed point is one minus one divided by r. At r = 2.5 this gives 0.6. Substitution gives 2.5 × 0.6 × 0.4 = 0.6.
Existence is not stability. A pencil balanced on its tip has an equilibrium in an ideal model but does not return after a small disturbance. To test attraction, perturb the state and watch the error rather than only checking the exact equilibrium.
Write f(x) for the next state produced from x. The letter f names the rule, so f(x) = r x times one minus x. A fixed point p must satisfy f(p) = p, with the same value on both sides. Moving everything to one side gives p times the quantity r minus r p minus one equal to zero. A product is zero when at least one factor is zero. This separates the possible solutions without assuming in advance that the state is positive.
One solution is p = 0. The other comes from r minus r p minus one equal to zero. Rearranging gives r p = r minus one, then p = 1 minus 1/r when r is nonzero. Dividing the original equation by p would lose the zero solution. That omission would matter even if zero is not the state you expect to observe. Fixed points describe all states that remain unchanged under the stipulated rule, not only the states that attract common starting values.
p denotes a dimensionless fixed state, not a probability in this lesson; r is the fixed map parameter.
The second formula requires nonzero r and must still pass the domain check; this is existence, not attraction.
| Factor set to zero | Candidate | Domain check |
|---|---|---|
| p | 0 | Always allowed |
| r − r p − 1 | 1 − 1/r | Positive for r > 1 |
| Both at r = 1 | 0 | One distinct state |
Consider a new parameter, r = 2.4. The nonzero solution is 7/12, approximately 0.583333. Substitution can stay in fractions. Multiplying 12/5 by 7/12 and then by 5/12 gives 7/12 again. This is an exact fixed point of the mathematical map. The rounded decimal 0.583333 is a nearby number, not the exact fraction. It is adequate for a short display, but treating its last digit as exact changes the starting state. Whether that difference shrinks is a separate calculation. The substitution establishes existence only, before any perturbation test.
An algebraic solution also has to belong to the allowed state interval. If r = 0.8, the expression 1 minus 1/r gives −0.25, outside the interval from zero to one. It solves the unrestricted algebraic equation but is not an admissible state here. At r = 1 the two formulas give the same state, zero, rather than two distinct fixed points. Counting formulas without checking their values would misdescribe this case. Existence therefore requires both solving the equation and checking the model's stated domain.
A useful quantity is the one-step change f(x) minus x. Factoring it gives x times the quantity r minus one minus r x. At r = 2.4 and an interior state below 7/12, this change is positive. Above 7/12 it is negative. The sign tells you the direction of that update relative to the present state. It does not tell you whether the new state stops before the fixed point or crosses it. That distinction requires comparing the output with p, not with x. Crossing and decreasing distance can therefore occur in the same update.
| Input at r = 2.4 | Next state | Next minus 7/12 |
|---|---|---|
| 0.58 | 0.58464 | +0.001306667 |
| 7/12 | 7/12 | 0 |
| 0.59 | 0.58056 | −0.002773333 |
For example, at r = 2.4 a state of 0.58 gives 0.58464 next. The change is positive, yet the new state is above 7/12. Starting at 0.59 gives 0.58056, below the fixed point. Both updates cross the fixed point while reducing the distance in these examples. A graph of the rule against the diagonal can reveal the direction of change. A graph of displacement from the fixed point reveals the crossing and contraction more directly. These two comparisons subtract different reference values for different purposes.
The local slope of the update curve measures its small-change response. At the nonzero fixed point that slope is 2 minus r. A slope with magnitude less than one shrinks sufficiently small errors. A negative slope also reverses the side of the error.
At r = 2.5, a small error is approximately multiplied by −0.5 at each update. At r = 3.2, the multiplier is −1.2, so nearby errors initially grow and alternate sides. This is a local approximation; large errors need the full nonlinear rule. In this lesson the lab starts 0.001 above the fixed point and compares the first eight actual errors with the multiplier approximation.
Let e be the signed error x minus p. Positive e means the state is above the fixed point, while negative e means it is below. Substitute x = p + e into the logistic rule and expand the product with ordinary algebra. The constant term is r p times one minus p, which equals p because p is fixed. After subtracting p, the remaining terms give the next error as r times one minus twice p times e, minus r e squared. This identity is exact.
For any fixed point p, f(p) = p and eₙ = xₙ − p. All quantities here are dimensionless.
At the nonzero fixed point, replacing p by 1 minus 1/r turns the coefficient of e into 2 minus r. Thus the exact error update is the linear contribution, 2 minus r times e, followed by the correction −r e squared. Squaring makes that correction nonnegative before its minus sign, regardless of the side of the initial error. The simple multiplier test drops this quadratic term. It is an approximation because it omits a known term, not because the model contains an unspecified random disturbance.
This substitution uses the nonzero branch. It is not the multiplier at the zero fixed point.
eₙ = xₙ − p is signed displacement from the nonzero fixed state; all values are dimensionless.
Signed error / initial 0.001
Update n, dimensionless
At r = 2.4, take e = 0.01 as a worked example distinct from the lab's smaller perturbation. The linear contribution is −0.004. The quadratic correction is −0.00024, so the exact next error is −0.00424. With e = −0.01, the linear contribution becomes +0.004, but the correction remains −0.00024. The next error is +0.00376. Equal starting distances on opposite sides need not produce equal ending distances. The multiplier approximation becomes symmetric only because it neglects that squared-error term. This asymmetry follows directly from multiplying the two state factors.
The linear term changes sign; the quadratic correction does not.
We can quantify when neglecting the correction is reasonable. Away from a zero multiplier, divide the magnitude r e squared by the magnitude of the linear contribution. The ratio is r times the absolute error divided by the absolute value of 2 minus r. At r = 2.4 it equals six times the absolute error. An error of 0.01 gives a correction six percent as large as the linear term. An error of 0.001 gives 0.6 percent. These percentages compare terms in one update, not accumulated forecast accuracy.
There is also a direct local contraction bound. The exact identity implies that the next absolute error is no larger than the current absolute error multiplied by the sum of the multiplier magnitude and r times the current absolute error. At r = 2.4, restricting the absolute error to at most 0.05 makes that sum at most 0.52. Every update then stays inside the same neighborhood and reduces the error. This establishes attraction within this stated neighborhood rather than merely observing a few shrinking values on a plot. The bound covers both signs of the initial displacement.
| Initial error | Linear term | Quadratic correction |
|---|---|---|
| +0.01 | −0.004 | −0.00024 |
| −0.01 | +0.004 | −0.00024 |
| +0.001 | −0.0004 | −0.0000024 |
The zero-multiplier case needs its own interpretation. At r = 2 the nonzero fixed point is 0.5, and the exact next error is −2 e squared. The first-order estimate predicts zero error, but the actual next error is usually small and negative rather than zero. Starting with an error of 0.01 gives −0.0002 next and −0.00000008 after another update. The error decreases faster near this point than a fixed nonzero multiplier would suggest. A vanished linear term makes the omitted term more important, not less important.
x is dimensionless and n counts updates. r is fixed within a run. We restrict 0 ≤ r ≤ 4 and 0 ≤ x ≤ 1.
Changing r restarts the run from the new fixed point plus 0.001, currently 0.601000; it does not continue the old run. The first sixty updates remain separate from the late sample.
This second plot magnifies the first eight displacements by 1,000. A plotted value of 1 means an error of 0.001. The scale adapts to include both traces. The approximation becomes less reliable as the displacement grows.
Nonzero fixed point: 0.6000. Local error multiplier: -0.50. Its magnitude is below one.
| Update | Late value |
|---|---|
| 893 | 0.600000 |
| 894 | 0.600000 |
| 895 | 0.600000 |
| 896 | 0.600000 |
| 897 | 0.600000 |
| 898 | 0.600000 |
| 899 | 0.600000 |
| 900 | 0.600000 |
| r | Fixed point | Multiplier | Local result |
|---|---|---|---|
| 2.5 | 0.6 | −0.5 | Alternating contraction |
| 3.2 | 0.6875 | −1.2 | Alternating expansion |
The lab's stability version starts each run at its current nonzero fixed point plus 0.001. Changing r therefore changes both the rule and the reference starting state. It is not a controlled experiment that keeps x fixed while varying only the parameter. The first plot shows sixty model updates, while the second isolates the first eight errors around the fixed point. Neither horizontal axis represents seconds or atmospheric days. Use the early error plot to test the local calculation, and the longer state trace to inspect where the run subsequently goes.
Select the attracting-point preset at r = 2.8, then use the slider to set r = 2.7. The multiplier becomes −0.7, so a linearized signed error alternates and retains seventy percent of its magnitude each update. With the lab's starting error of 0.001, the exact first error is −0.0007027. The next is approximately +0.000490557. These are displacements from the new fixed point, approximately 0.629630, rather than values of the state itself. Subtracting the reference point prevents the baseline height from obscuring the small changes.
Linear error / initial error
Update n, dimensionless
The plotted vertical quantity is error multiplied by one thousand. An actual displacement of −0.0007027 therefore appears near −0.7027, not near −0.0007027. The linearized trace starts at one on this scale because the initial error is 0.001. Both traces use the same axes within a selected run, so their separation is meaningful. The vertical bounds can change when you select another parameter. A taller-looking wiggle in a different run is not enough to establish a larger physical or mathematical error without reading the tick values.
Now compare r = 2.7 with r = 2.95 using the slider. The multiplier magnitudes are 0.7 and 0.95. Both are below one, but the second removes a much smaller fraction of the error at each update. In the linear approximation, after eight updates the remaining fractions are about 0.057648 and 0.663420. Those fractions divide by the initial error, not by the fixed-point state. Slow decay near a stability boundary can look almost stationary over a short record even when the strict local test still predicts attraction. Stability alone does not specify a useful settling horizon.
The linear approximation does not recalculate its multiplier along the moving orbit. It keeps the slope at the fixed point and raises that number to successive integer powers. The actual nonlinear trace instead applies the full logistic rule at every step. Their agreement near the initial state tests the usefulness of this local approximation over the displayed interval. Disagreement later does not imply that the implementation has added noise. It can arise because the neglected squared-error contribution changes each step and its effects accumulate through later applications of the rule.
| At r = 2.7 | Actual error | Error × 1000 |
|---|---|---|
| Update 0 | +0.001 | +1 |
| Update 1 | −0.000702700 | −0.702700 |
| Update 2 | +0.000490557 | +0.490557 |
The late table has a different purpose from the magnified error plot. It reports the last eight states from a separate calculation reaching update 900. Near-equal printed values support a settling hypothesis, but the displayed rounding can hide small differences. At a setting where the fixed point repels, the late states can alternate among other values even though the early errors move away from p. The table does not extend the local straight-line approximation to 900 updates. It samples the nonlinear recurrence, with the same start rule used for that selected parameter.
For 1 < r < 3 the nonzero fixed point is locally attracting. Exactly at a multiplier magnitude of one, this first-order test is inconclusive; higher-order behavior matters. Do not extend a strict inequality to its boundary without analysis.
A growing displacement from a fixed point need not grow forever. It may approach a stable cycle elsewhere in the bounded interval. Instability of one equilibrium is not proof of chaos throughout the system.
The multiplier test uses a strict comparison with one because its proof needs a margin for the omitted nonlinear term. If the multiplier magnitude is 0.7, sufficiently small quadratic corrections cannot overturn contraction. If its magnitude is exactly one, there is no such margin. This does not mean the state necessarily repels or necessarily attracts. It means the linear term alone cannot settle the question. The full error equation can still provide an answer, sometimes by examining more than one update at a time rather than changing the inequality by convention.
At r = 3, write the current error as e relative to p = 2/3. One update gives −e minus 3 e squared. Applying that error rule again and expanding gives the two-update error e minus 18 e cubed minus 27 e to the fourth power. For e = 0.001 this equals approximately 0.000999981973. The return error is slightly smaller even though the first error, −0.001003, has slightly larger magnitude. This calculation explains why a one-step magnitude test at the boundary misses behavior visible over two steps.
Signed error / initial 0.001
Update n, dimensionless
For sufficiently small errors at r = 3, divide that two-update expression by e when e is nonzero. The ratio is one minus e squared times the quantity 18 plus 27 e. Near zero, the quantity in that last pair is positive and the ratio is positive but below one. Thus two-step errors contract locally, though the ratio approaches one as the error shrinks. This is much slower than contraction by a fixed factor comfortably below one. The conclusion comes from the higher-order terms, not from the inconclusive linear test.
e is displacement from the fixed point 2/3. The first-order factor −1 alone is inconclusive.
The leading correction after two updates explains the local contraction discussed here. It is not a global convergence claim.
A local attraction statement still leaves other starting states unexamined. At r = 2.4, zero stays zero forever, and one maps immediately to zero. Neither approaches the nonzero fixed point. Those exact endpoint cases already prevent a claim that every state in the closed unit interval converges to 7/12. A basin of attraction means the set of starts that do converge to a specified attracting state or cycle. Our explicit neighborhood bound proves membership for nearby starts, not a complete description of that set throughout the interval.
Changing the parameter can move the target even when local attraction remains. At r = 2.4 the nonzero fixed point is 7/12. At r = 2.7 it is 17/27. Their difference is 5/108, approximately 0.046296. A small initial-state error under one fixed rule is therefore not the same experiment as using a different parameter. The first asks how neighboring trajectories evolve toward the same target. The second changes both the target and its local error multiplier. Comparing forecasts requires specifying which source of difference the calculation represents.
| At r = 2.4 | Next state | Consequence |
|---|---|---|
| Start 0 | 0 | Remains at zero |
| Start 1 | 0 | Then remains at zero |
| Start 7/12 | 7/12 | Fixed nonzero state |
| |x − 7/12| ≤ 0.05 | |next error| ≤ 0.52 |error| | Proved local contraction bound |
This distinction also limits applications to physical systems. An equilibrium calculation assumes that the rule and its parameters remain appropriate over the interval being considered. A stable fixed point under a misspecified rule can attract numerical trajectories while missing the behavior of the real system. The logistic calculation establishes a conditional mathematical result, including a neighborhood where perturbations shrink. It does not estimate observational error or demonstrate that atmospheric states converge to an equilibrium. To make that further claim would require a justified atmospheric model and evidence about the relevant states and forcing.
At r = 3.2, exactly x = 0.6875 stays fixed in exact arithmetic. Why can nearby runs leave it?
Compare the error multiplier with one in magnitude.
| Update | Approximate error |
|---|---|
| 0 | 0.001 |
| 1 | −0.0012 |
| 2 | 0.00144 |
The multiplier is −1.2. Small deviations reverse sign and grow initially. Exact equality is a special trajectory, not evidence that nearby initial states converge to it.
For r = 1.5, find the nonzero fixed point and predict whether small errors alternate.
Use one minus 1/r and 2 minus r.
| Multiplier | Magnitude | Sign | Local behavior |
|---|---|---|---|
| +0.5 | Below one | Positive | Contract on the same side |
| −0.5 | Below one | Negative | Contract and alternate |
The fixed point is one third. The multiplier is +0.5, so small errors contract without reversing side at first order. This differs from the alternating contraction at r = 2.5.
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.
Recurrence, fixed points, period doubling, and periodic windows. Selected explanatory passages reviewed; the plots here are generated independently.