Connect a three-variable state to a phase-space trajectory.
By the end: Interpret the Lorenz model without mistaking its coordinates for a weather map.
The Lorenz system gives rates of change for three quantities x, y, and z. It arose as a simplified convection model. In its original interpretation, these describe circulation and temperature-pattern amplitudes, not longitude, latitude, and altitude. Time and coordinates here are dimensionless.
The model contains products xz and xy, so the variables affect each other nonlinearly. We use the classical parameters sigma = 10, rho = 28, beta = 8/3. A fourth-order Runge–Kutta method approximates the solution at time steps of 0.01, up to thirty model-time units.
A rate tells you how much a quantity would change per unit time if its present rate persisted. It does not tell you the quantity itself. At the fixed parameters, the x, y, and z rates are respectively 10(y − x), x(28 − z) − y, and xy − (8/3)z. For the Lorenz system, substitute the current three numbers into all three rate formulas before advancing anything. The result is another triple, called the tendency. Multiplying that triple by a short time interval gives a first approximation to the change in state. This uses the same algebra as distance equals speed times time, except that the speeds now depend on the evolving state.
Consider the constructed state x = 3, y = 4, z = 12 with the lesson's fixed parameters. The x rate is 10 times the difference between four and three, giving 10. The y rate is three times sixteen minus four, giving 44. The z rate is twelve minus thirty-two, giving −20. Thus x and y initially rise while z falls. These are dimensionless amplitudes per dimensionless model-time unit. They are neither wind speeds nor temperature changes in degrees per hour, even though convection motivated the original equations.
Dots mean rates per dimensionless model-time unit. All three rates use the same constructed state (x,y,z) = (3,4,12); coordinates are dimensionless amplitudes.
These are instantaneous tendencies, not full-step changes or atmospheric rates in physical units.
| Coordinate | Substitution | Rate |
|---|---|---|
| x | 10 × (4 − 3) | 10 |
| y | 3 × (28 − 12) − 4 | 44 |
| z | 3 × 4 − (8/3) × 12 | −20 |
Over an interval of 0.01 model-time units, freezing those rates would predict changes of 0.10, 0.44, and −0.20. Adding them to the starting coordinates gives 3.10, 4.44, and 11.80. This is a forward Euler estimate, introduced here only to explain what a tendency does. The displayed Lorenz lab instead uses fourth-order Runge–Kutta. The frozen-rate estimate cannot be an exact update because the intermediate state changes the rates during the interval. Its simplicity helps separate a differential equation from an algorithm used to approximate its solution.
The dependence on other coordinates explains why reading one rate alone can mislead. In the constructed state, y grows much faster than x. That changes the difference y minus x, which then changes the x rate. Meanwhile a falling z increases the factor twenty-eight minus z in the y equation. These effects operate together, rather than as three independent decay or growth rules. Their initial signs describe a small neighborhood of this state. They do not establish the signs throughout a complete circuit around either lobe of the computed trajectory.
To see the feedback without integration, hold x = 3 and z = 12 fixed and compare several possible y values. Increasing y raises the x rate and the z rate but lowers the y rate directly. The same change therefore has different effects on different equations. The nearby plot is a parameter scan through constructed states, not one evolving run. Its straight connections are exact for these rate formulas with x and z held fixed. A real trajectory does not keep those two coordinates fixed while varying y.
Rate per model-time unit
Constructed y, dimensionless; x = 3, z = 12
The products explain the nonlinearity more precisely than the shape of a plotted curve does. Doubling both x and y multiplies xy by four, while doubling z only doubles the subtraction involving z. Therefore doubling every coordinate does not generally double every tendency. Linear scaling of a state cannot substitute for recalculating its rates. The constants still remain unchanged in this comparison. In the actual lab, the available buttons change the initial x difference only, not the parameters, integration step, or interpretation of the coordinates as dimensionless amplitudes.
The x–z plot projects the three-dimensional state onto two axes. Nearby drawn points can differ in the hidden y coordinate. An apparent crossing in this projection is not a crossing of identical complete states with different deterministic futures.
A typical computed trajectory circles two regions without settling to a simple repeated path. The familiar two-lobed shape is a phase-space structure, not the shape of a storm. The full trace stays visible; this comparison does not need decorative looping animation.
A phase portrait removes time from the horizontal axis, but the trajectory still has a time order. Consecutive numerical states provide that order. The calculation stores their order. The unarrowed projected line shows connectivity but does not by itself identify the direction of travel. The line does not mean that the exact continuous solution follows a straight segment. Even a detailed projected portrait can conceal how quickly a state traverses a region. Equal distances along the drawing need not take equal model-time intervals, and a thick band may contain several passages separated by substantial model time.
There is a useful algebraic symmetry in the Lorenz equations. Reverse the signs of both x and y while keeping z unchanged. The x and y rates reverse signs, and the z rate stays the same because the product xy keeps its sign. Thus a whole solution has a partner obtained by reflecting its x and y coordinates. For example, the constructed states 3, 4, 12 and −3, −4, 12 have related tendencies. This explains a symmetry of the equations without claiming that a short plotted trajectory must spend equal time on both sides.
| Quantity | State 3, 4, 12 | State −3, −4, 12 |
|---|---|---|
| x rate | 10 | −10 |
| y rate | 44 | −44 |
| z rate | −20 | −20 |
A symmetric rule need not produce a symmetric finite sample. A run can spend longer circling one region before switching to the other, and the chosen start can influence the early portion. Counting points in the two lobes over thirty model-time units therefore describes this computed sample. It does not prove equal long-run occupancy, nor does it establish the chance of a switch at the next instant. That would require a defined distribution of starting states and evidence about how trajectories sample the relevant region over much longer intervals.
Time plots answer a different question from phase portraits. Plotting x against model time shows when sign changes occur and whether two calculations remain close in that coordinate. Plotting z against x instead shows how those quantities vary together, with the time labels omitted. One cannot translate a horizontal distance in the second plot into elapsed time. The lab makes both views available simultaneously because they retain different information. Neither view alone displays all three coordinates, so the full-state separation in the table supplies a separate numerical comparison.
Even distance requires specifying which coordinates enter the calculation. For a constructed displacement of 0.3 in x, 0.4 in y, and 1.2 in z, square the three differences and add them. The sum is 1.69, whose square root is 1.3. Using only x and z gives the square root of 1.53, about 1.237. The projection underestimates this full-state distance because it discards a nonnegative squared contribution. These dimensionless distances are appropriate for these model amplitudes, not a general recipe for combining unlike physical units in atmospheric verification.
d is Euclidean full-state distance. The constructed coordinate gaps are Δx = 0.3, Δy = 0.4, Δz = 1.2, all dimensionless.
The projection omits the nonnegative y contribution 0.16; it cannot reconstruct the full-state distance.
| Contribution | Squared difference |
|---|---|
| x difference 0.3 | 0.09 |
| y difference 0.4 | 0.16 |
| z difference 1.2 | 1.44 |
| Sum across all coordinates | 1.69 |
| Square root of sum | 1.30 |
In a weather state, temperature, pressure, and velocity have different units and natural scales. Adding their raw squared differences would make the answer depend on arbitrary unit choices. A meaningful weather comparison needs specified scaling or a physically justified measure. The Lorenz lab avoids that complication by showing dimensionless coordinates and explicitly naming the Euclidean full-state norm. Still, a small x difference alone cannot establish a small norm. The selected full-state rows are useful checks at particular times, not a continuous guarantee between the times printed in the table.
Reference starts at (1, 1, 1). Comparison adds 0.000001 to x only. Each choice recalculates the comparison with the same step of 0.01; no random forcing is used.
| Model time | Reference x | Perturbed x | Full-state separation |
|---|---|---|---|
| 0 | 1.0000 | 1.0000 | 1.000e-6 |
| 5 | -6.5120 | -6.5120 | 1.245e-6 |
| 10 | -4.9028 | -4.9028 | 1.758e-6 |
| 20 | 13.4729 | 13.4725 | 1.069e-3 |
| 30 | 9.8669 | -16.1543 | 3.361e+1 |
| Variable | Rate |
|---|---|
| x | 10(y − x) |
| y | x(28 − z) − y |
| z | xy − (8/3)z |
Runge–Kutta improves on a frozen-rate update by evaluating tendencies at trial intermediate states. Its four evaluations estimate how rates change during one interval. The final weighted combination uses weights one, two, two, and one, divided by six. Those weights do not average four independent physical measurements. They combine predictions from the same equations. Nor are the trial states four successive completed updates. Only the final combination advances the numerical state by the chosen step. Repeating that calculation produces the sampled trajectory shown by the Lorenz lab.
For a simple comparison that can be checked without three-variable arithmetic, consider a separate dimensionless equation whose rate is minus its current amount. Start with amount two and take a step of 0.2 model-time units. Fourth-order Runge–Kutta multiplies the amount by 1 − h + h²/2 − h³/6 + h⁴/24, where h is 0.2. The multiplier is about 0.8187333. This special polynomial comes from applying the four stages to this linear equation. It is not the general update formula for the nonlinear Lorenz system.
Dimensionless amount
Dimensionless model time; initial amount 2
The exact amount in this separate decay example is two times exp of minus model time. Here exp denotes the exponential function, whose constant fractional decay preserves positivity. After one step the numerical amount is about 1.6374667, while the exact value is about 1.6374615. Their difference is small but not zero. A smooth line and many printed digits would not remove it. This example deliberately uses another initial amount and another equation than the Lorenz lab, so its numerical agreement cannot certify the accuracy of the displayed chaotic trajectory.
Only for the separate scalar decay problem, with dimensionless h = 0.2: R₂ is a partial polynomial and R₄ the full RK4 retention, not Lorenz update formulas.
Exact amount is 2 exp(−0.2) ≈ 1.6374615. This scalar check does not validate the thirty-unit Lorenz trajectory.
A step refinement comparison asks whether reducing the step changes the answer at the same endpoint. For the separate decay example, integrate to model time one using steps of 0.2, then repeat with steps of 0.1. The errors relative to two times exp of minus one are about 0.00001159 and 0.000000667, respectively. The smaller step improves this example. The comparison holds total model time fixed, rather than holding the number of updates fixed. Otherwise the two calculations would answer different prediction questions, even if they used identical algorithms.
The word fourth-order describes how global error scales as the step becomes small under suitable smoothness and fixed-interval assumptions. It does not promise four correct decimal places. For this decay comparison, halving the step reduces error by roughly seventeen, close to the factor sixteen expected in the small-step regime. Nonlinear error growth, changing trajectories, and floating-point arithmetic complicate longer Lorenz comparisons. The lab fixes its step at 0.01. The decay calculations here explain a numerical method; they are static examples, not extra controls hidden in the interface.
| Step and count | Amount at time 1 | Absolute error |
|---|---|---|
| 0.2, five steps | 0.7357704763 | 0.0000115939 |
| 0.1, ten steps | 0.7357595488 | 0.0000006665 |
| Exact exponential | 0.7357588823 | 0 |
Roundoff and truncation have different causes. Roundoff comes from representing numbers with finitely many digits. Truncation comes from replacing continuous evolution by a finite approximation, even if every arithmetic operation were exact. Changing a starting coordinate tests initial-state sensitivity, while changing the integration step tests part of the numerical approximation. The lab buttons perform the first comparison only. With Identical selected, overlapping computed curves test repeatability of the same calculation. They do not show that roundoff vanished or that the computed path is the exact continuous solution over thirty model-time units.
An attractor can draw trajectories toward a set while trajectories within that set separate from each other. This resolves the apparent conflict between long-run bounded structure and short-run sensitivity. Attraction to a region is not attraction to one exact history.
The second trace begins with x larger by a selected small amount. Finite-step numerical separation illustrates sensitivity but does not prove the Lorenz attractor exists. Special initial conditions, such as equilibria, need not trace the familiar pattern. Never claim every possible trajectory is chaotic.
Attraction to a set concerns where trajectories eventually lie, while sensitivity concerns how neighboring trajectories move relative to each other. Those statements involve different distances. Distance to a set means distance to the nearest allowed point in that set. Distance between two states compares those particular points. Two trajectories can each approach a restricted region while their mutual distance grows inside it. A familiar two-lobed drawing supports the idea of restricted geometry, but a finite numerical picture does not establish the limiting set or its mathematical attraction properties.
A simple exactly solvable comparison makes the distinction concrete without claiming another chaotic system. Imagine radius r shrinking toward one according to r = 1 + exp of minus time. Two points share this radius, but their angles differ by 0.1 radians. Let both angles advance at the same constant rate. Their distance to the unit circle decreases, while their mutual distance is twice r times the sine of 0.05. The points approach a set without approaching the same point. This regular example separates two meanings of attraction before adding sensitivity. Radians measure an angle by the arc length it cuts from a unit circle. The gap 0.1 radians is about 5.73 degrees. Bisecting the straight segment between the points forms two right triangles with hypotenuse r and angle 0.05. Sine means opposite side divided by hypotenuse. Each half-segment therefore has length r times sine of 0.05; doubling gives the stated separation.
Dimensionless distance
Dimensionless time in the constructed circle example
In that comparison, the two points' separation decreases toward about 0.1000 rather than zero. The circle attracts their radii, but their angular offset persists. It is therefore wrong to infer synchronized histories merely from attraction to a common set. The Lorenz model adds irregular motion and nearby-state separation that this circle example lacks. The circle is a mathematical counterexample to an inference, not a surrogate proof of Lorenz chaos. Its time is dimensionless, its radius is dimensionless, and none of its constants comes from observed atmospheric behavior.
A bounded system also prevents a fixed exponential separation law from continuing forever. Once two states are separated by the size of the accessible region, further unconstrained exponential growth would leave that region. Local growth can slow, reverse, or depend on direction. Thus the absolute-distance trace need not increase at every sampled time. A dip in distance does not by itself refute sensitivity, just as an early increase does not prove sustained chaotic dynamics. A quantitative growth estimate needs a defined interval, distance measure, and treatment of separations near the system's overall scale.
The lab's One millionth and One thousandth choices perturb x alone from the same reference state. Comparing their full-state distances at the printed times asks how those particular perturbations evolve under the same algorithm. It does not sample all directions or all initial states. Choosing Identical first provides a useful control for that interpretation. Then change only the initial difference and retain the fixed thirty-unit horizon. The resulting comparison demonstrates state-dependent forecast disagreement in this teaching model without assigning atmospheric days to the horizontal axis or probabilities to the two runs.
| Comparison | Concrete calculation | What it retains |
|---|---|---|
| At time 0 | x gap = selected delta | Initial perturbation only |
| At a printed time | Square, sum, root all gaps | Full-state separation |
| In the x time plot | Compare x values only | Time, one coordinate |
| In the x–z portrait | Plot x and z pairs | Geometry, two coordinates |
This model's weather relevance is a mechanism, not a weather forecast. Coupled deterministic rates can turn incomplete initial knowledge into substantial later disagreement while preserving recognizable state-space structure. To predict an actual atmospheric variable, one would also need a much larger physical state, an observation-based initial estimate, and a tested numerical model. The selected secondary Lorenz reference supplies equations and context, not operational forecast validation. Neither its familiar geometry nor these added calculations establishes a calibrated forecast horizon for a location, season, or particular kind of hazardous event.
Two lines cross in an x–z projection. Must the deterministic model be inconsistent?
One state coordinate is hidden.
| State | x | y | z | Rate of x |
|---|---|---|---|---|
| A | 2 | 3 | 10 | 10 |
| B | 2 | −1 | 10 | −30 |
No. They can have equal x and z but different y. Even along one trajectory, a projected crossing need not be an exact repeat of the full state. Uniqueness concerns all state coordinates.
At x = y = z = 0, what are the three rates? Must this run visit both lobes?
Substitute zero into every product and difference.
| Rate | Substitution | Value |
|---|---|---|
| x rate | 10(0 − 0) | 0 |
| y rate | 0(28 − 0) − 0 | 0 |
| z rate | 0 × 0 − (8/3) × 0 | 0 |
All rates are zero, so the origin remains fixed in exact arithmetic. It need not visit either lobe. At the chosen parameters it is unstable, which is different from saying it is not a solution.
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 reference for the three-variable system and classical parameters. The course uses a numerical illustration, not a proof or operational atmospheric model.
Secondary overview of deterministic dynamics and sensitive dependence. Not a proof that every irregular numerical trace is chaotic.