Calculator Inputs
Use classic values or try new parameters to explore sensitivity and chaotic structure.
Example Data Table
| Scenario | σ | ρ | β | Initial State | dt | Steps | Typical Behavior |
|---|---|---|---|---|---|---|---|
| Classic chaotic | 10 | 28 | 2.6667 | (1, 1, 1) | 0.01 | 5000 | Butterfly-shaped chaotic orbit |
| Lower rho test | 10 | 18 | 2.6667 | (0.1, 0, 0) | 0.01 | 4000 | Reduced spread and calmer motion |
| Energetic case | 14 | 35 | 3 | (2, 3, 4) | 0.005 | 8000 | Wider excursions and stronger sensitivity |
Formula Used
The page integrates the Lorenz system numerically using a fourth-order Runge–Kutta method. This is more accurate than a basic Euler step for the same time increment.
Divergence is computed as −(σ + 1 + β). Because it is negative for common parameter choices, trajectories fold inward even while remaining highly sensitive to starting conditions.
Path length is estimated by summing the Euclidean distance between successive analysed points. Average speed equals path length divided by the analysed time window.
How to Use This Calculator
- Enter sigma, rho, and beta for the Lorenz system.
- Choose initial x, y, and z coordinates.
- Set the time step and total integration steps.
- Use transient skip to ignore startup behavior.
- Set graph point limit for faster plotting.
- Press Calculate Lorenz Attractor.
- Review summary cards, the trajectory graph, and the table preview.
- Download the analysed dataset as CSV or export a PDF report.
FAQs
1) What does this calculator measure?
It numerically simulates the Lorenz system and summarizes the resulting trajectory. You can inspect ranges, final state, path length, divergence, centroid, average radius, and a 3D curve showing how the system evolves over time.
2) Why is the Lorenz system important?
It is a classic example of deterministic chaos. Tiny changes in starting values can create very different paths, even though the equations are fixed and simple. That makes it useful for teaching nonlinear dynamics and sensitive dependence.
3) What are common default values?
A famous parameter set is σ = 10, ρ = 28, and β = 8/3. With a small time step and a modest initial point such as (1, 1, 1), the trajectory often forms the well-known butterfly-shaped attractor.
4) Why should I use a transient skip?
Initial steps may reflect startup effects rather than long-run attractor structure. Skipping early points helps focus the analysis on stabilized chaotic motion, especially when you want cleaner summary statistics or a tidier graph.
5) What does divergence tell me here?
For the Lorenz equations, divergence equals −(σ + 1 + β). A negative value means volumes contract on average in phase space, which helps explain why trajectories fold into a bounded attractor instead of escaping indefinitely.
6) Why use RK4 instead of a simpler method?
Fourth-order Runge–Kutta usually gives better accuracy than Euler integration for the same step size. That matters in chaotic systems because numerical error can grow quickly and distort the trajectory if the solver is too crude.
7) Can I compare two parameter sets?
Yes. Run the calculator once, note the summaries, then change sigma, rho, beta, or initial conditions and calculate again. Comparing ranges, final states, and graph shape quickly reveals how sensitive the system is.
8) What export options are included?
The page includes CSV export for the full analysed dataset and PDF export for a compact report. CSV is better for detailed external analysis, while PDF is useful for sharing a summary snapshot.