Reasoning
Read the definitions, follow the arguments and examine the conditions of each result.
Background: Derivatives, Vectors, Initial-value problems. Start with the model for a brief introduction, follow the guided exploration for a closer look, or work through the arguments and experiment for a detailed study.
What you can learn
- Interpret a deterministic differential equation and a numerical trajectory separately.
- Compare initial perturbations at a fixed integration step.
- Use short-time step refinement to examine numerical reliability.
Reading step 1 of 5: Specify the Lorenz model
Step 1 · Derivation · Selected
Specify the Lorenz model
The Lorenz system is a deterministic set of three coupled differential equations. Given an initial state and fixed parameters, it specifies a mathematical evolution; there is no random term in the model used here. The familiar trajectory is a path through a space of state variables, not the flight path of an air particle.
The default parameters are σ = 10, ρ = 28 and β = 8/3, with starting state (1,1,1). Lorenz derived the system from a strongly reduced convection model. This exhibit studies that finite-dimensional model and its numerical solution, without presenting the results as an actual weather prediction.
Derivation. Three coupled differential equations define the Lorenz initial-value problem used here.
Conditions and assumptions
- The displayed parameters and initial state are fixed for each run.
- The coordinates and time are model variables.
Sources: [M7] Deterministic Nonperiodic Flow
Step 2 · Derivation
Find exact equilibrium checks
At an equilibrium all three derivatives vanish. The first equation gives y = x when σ is positive. The second becomes x(ρ − z − 1) = 0. If x is zero, the third equation forces z to zero when β is positive, giving the origin.
Otherwise z = ρ − 1, and the third equation gives x² = β(ρ − 1). For ρ > 1 this produces two states with matching signs for x and y. Substituting these expressions directly into the vector field is an independent benchmark. Finding equilibria does not establish their stability or classify every other trajectory.
Derivation. The origin is an equilibrium; when ρ > 1 there are two further equilibria.
Conditions and assumptions
- σ > 0 and β > 0; ρ > 1 for the nonzero equilibria.
- σ > 0 and β > 0.
- Matching signs are used for x and y.
- ρ > 1 for the nonzero pair.
Sources: [M7] Deterministic Nonperiodic Flow
Step 3 · Numerical experiment
Distinguish two comparisons
For the sensitivity comparison, both trajectories use the same parameters, algorithm and integration step. Their initial states differ only by ε in the first coordinate. The distance curve records the Euclidean norm of their separation at the same computed times. With ε = 0, the same calculation should produce identical trajectories.
For the discretization comparison, hold the initial state fixed and change the integration step from h to h/2. Compare only shared time locations. These two experiments isolate different changes in the calculation; a single separating pair cannot tell them apart. Neither finite numerical comparison proves every rigorous property associated with chaos.
Numerical experiment. The distance between two computed trajectories measures separation for this particular run.
Conditions and assumptions
- Both trajectories use the same solver, step and parameter values.
- Only the first initial coordinate differs by ε.
- A finite trajectory comparison is not a proof of all properties of chaos.
Sources: [M7] Deterministic Nonperiodic Flow
Step 4 · Derivation
Check the time integrator
The implementation uses fixed-step fourth-order Runge–Kutta with double-precision arithmetic. Each update combines four evaluations of the differential equation at prescribed intermediate states. Display playback consumes the completed samples; monitor refresh rate is not a numerical step size.
An independent test integrates y′ = y from y(0) = 1 and compares against exp(t). Short-window step refinement then examines the Lorenz calculation before long-time separation dominates the comparison. A smaller step is not a promise of indefinitely accurate pointwise prediction. Lorenz’s original paper used a different numerical procedure; this browser solver is a documented project choice.
Sources: [M7] Deterministic Nonperiodic Flow; [M12] Runge–Kutta methods
Step 5 · Numerical experiment
Report a finite observation
The threshold measurement is the first sampled time at which the recorded separation exceeds the chosen value. Its resolution depends on the integration step. If no sample crosses within the observation window, the result remains not observed; the window endpoint is not substituted as a crossing time.
A zero distance is retained as zero and treated separately on logarithmic plots. The study stores the full solver settings and arrays, even if the scene displays fewer points. No maximum Lyapunov exponent is estimated here, and a short-time slope or distance ratio is not labeled as one. Numerical results remain subject to finite precision and finite observation time.
Sources: [M7] Deterministic Nonperiodic Flow
References and review status
Project attribution: BIPH Math Society. The exposition was prepared with tool assistance.
Review status: Not reviewed. Independent review by a teacher or society member has not yet been recorded. Numerical checks do not replace mathematical review.
[M7] Deterministic Nonperiodic Flow
Edward N. Lorenz
Journal of the Atmospheric Sciences 20 (1963), 130–141; equations and equilibrium analysis on p. 135, default parameter choices and original integration method on p. 136. The project’s fixed-step RK4 solver is its own documented choice and differs from the original double-approximation procedure. Source checked 16 September 2026.
[M12] Runge–Kutta methods
Richard Fitzpatrick
University of Texas at Austin computational physics course, 29 March 2006. Equations (25)–(29) give the four classical RK4 stages and local remainder. The project applies the vector form to its Lorenz system; the model paper is a separate source. Source checked 16 September 2026.