Explore chaotic flow through adjustable Lorenz parameters. Run Euler or RK4 steps with custom timestep. View phase behavior, then download tables in seconds easily.
The Lorenz system is a three-variable nonlinear model often used to study chaos:
This solver advances the state over time using Euler or fourth‑order Runge–Kutta (RK4) integration.
| σ | ρ | β | x₀ | y₀ | z₀ | dt | steps | method |
|---|---|---|---|---|---|---|---|---|
| 10 | 28 | 2.6667 | 1 | 1 | 1 | 0.01 | 5000 | RK4 |
| 14 | 35 | 3.0000 | 0.1 | 0 | 0 | 0.005 | 12000 | RK4 |
| 10 | 24 | 2.6667 | 2 | 2 | 2 | 0.01 | 4000 | Euler |
The Lorenz system is a compact model that produces rich, chaotic motion from three coupled ordinary differential equations. This calculator numerically integrates the state over time so you can study sensitivity to initial conditions, parameter tuning, and long‑run bounded behavior using reproducible settings.
The solver advances x(t), y(t), and z(t) under the standard Lorenz flow. With σ controlling coupling, ρ acting as forcing, and β setting dissipation, small changes can shift the trajectory from steady convergence to oscillations and chaotic wandering.
A widely used benchmark set is σ = 10, ρ = 28, and β = 8/3. With modest initial values such as (1, 1, 1), many simulations exhibit the classic “butterfly” attractor. Reducing ρ toward the low‑20s often weakens chaotic structure, while increasing ρ can enlarge the explored range.
Euler integration is fast and simple, but it can accumulate error quickly in nonlinear chaotic systems. Fourth‑order Runge–Kutta (RK4) evaluates intermediate slopes to improve accuracy per step. For comparable settings, RK4 typically preserves bounded dynamics with fewer artifacts at practical time steps.
The time step dt is the most important numerical control. Smaller dt reduces truncation error and improves stability, but increases runtime. Values around 0.01 are common for exploratory runs; reducing to 0.005 or 0.001 can help when you increase steps or push parameters into more extreme regimes.
Total steps define the simulation horizon T = steps × dt. Because saving every step can create very large tables, the output stride stores one row every N steps. For example, 50,000 steps with stride 10 yields about 5,001 saved rows, balancing detail with lightweight exports.
The results panel reports the final state and the observed minima and maxima for x, y, and z over the run. These ranges are useful quick diagnostics: exploding ranges can indicate an unstable dt or method choice, while tight ranges may reflect convergence or a parameter set outside chaotic windows.
After you submit inputs, the trajectory table appears immediately above the form, ready for review. Use the CSV export to load data into spreadsheets or scientific tools, and use the PDF export for reporting. Keeping stride consistent makes comparisons across multiple parameter sweeps straightforward.
It numerically integrates the Lorenz differential equations to produce a time‑ordered trajectory of (t, x, y, z) using your parameters, initial conditions, time step, and selected method.
RK4 is recommended for accuracy and better stability at practical dt values. Euler can be useful for quick demonstrations, but it may distort chaotic dynamics unless dt is very small.
dt controls truncation error. In chaotic systems, small numerical differences grow rapidly, so trajectories diverge over time. Smaller dt generally improves fidelity and reduces instability artifacts.
A common starting set is σ=10, ρ=28, β=8/3 with initial (1,1,1). This often produces the classic attractor shape while keeping the run stable with dt around 0.01.
Increase output stride to save fewer rows while keeping the same dt and steps. You’ll still capture overall structure, and exports remain fast and manageable.
This can happen when dt is too large, Euler is used at aggressive settings, or parameters push the system numerically unstable. Try RK4, reduce dt, and verify your parameter magnitudes.
Yes. Keep dt, steps, and stride consistent, then vary one factor such as ρ or initial conditions. Use the CSV export to overlay trajectories or compute statistics across runs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.