Calculator Inputs
Example Data Table
| System | Parameter A | Parameter B | Parameter C | Initial State | dt | Steps | Projection |
|---|---|---|---|---|---|---|---|
| Lorenz | 10 | 28 | 2.6667 | (0.1, 0, 0) | 0.01 | 5000 | X-Y |
| Rössler | 0.2 | 0.2 | 5.7 | (0.1, 0, 0) | 0.02 | 5000 | X-Z |
| Duffing | 0.2 | -1 | 0.3 | (0.1, 0, 0) | 0.02 | 6000 | 3D |
Formula Used
dX/dt = F(X, p), where X = [x, y, z] and p contains model parameters.
The calculator uses the fourth-order Runge–Kutta method:
Xn+1 = Xn + (dt/6)(k1 + 2k2 + 2k3 + k4)
dx/dt = σ(y − x), dy/dt = x(ρ − z) − y, dz/dt = xy − βz
dx/dt = −y − z, dy/dt = x + ay, dz/dt = b + z(x − c)
dx/dt = y, dy/dt = −δy − αx − βx³ + γcos(phase), dphase/dt = ω
The phase portrait is built by plotting retained trajectory coordinates after transient removal. This reveals attractors, loops, dense folds, and sensitivity patterns.
How to Use This Calculator
- Select a chaotic model from the system list.
- Enter parameter values for the chosen model.
- Set initial x, y, and z conditions.
- Choose time step, total steps, and discarded transient steps.
- Select a phase projection such as X-Y or 3D.
- Press Generate Phase Portrait to compute results.
- Review the plotted attractor, summary metrics, and sample points.
- Use the CSV or PDF buttons to export the output.
Frequently Asked Questions
1. What does a chaos phase portrait show?
It plots the evolving state of a nonlinear system in phase space. The shape reveals attractors, periodic loops, divergence regions, and sensitive dependence on initial conditions.
2. Why remove transient steps?
Early motion may reflect startup behavior rather than long-term dynamics. Removing transient steps helps isolate the stable attractor structure for clearer visual interpretation.
3. What is the best time step?
A smaller time step usually improves accuracy, but increases computation time. Start with the default value, then reduce it if the portrait looks jagged or unstable.
4. Why do two nearby starting points diverge?
That is a hallmark of chaos. Small differences in initial conditions can grow quickly, producing very different trajectories even when the governing equations stay unchanged.
5. When should I use 3D projection?
Use 3D when the full state geometry matters, especially for Lorenz and Rössler systems. Use 2D projections when you want a simpler view of folded structures.
6. What does normalization do?
Normalization rescales retained coordinates into comparable ranges. It is useful when one axis dominates the others and hides structural details in the portrait.
7. Can this calculator prove chaos?
No. It helps visualize chaotic behavior and estimate ranges, but formal proof may require Lyapunov exponents, bifurcation analysis, Poincaré sections, or other tests.
8. Which model should beginners start with?
The Lorenz system is a good starting point because its attractor is familiar and visually distinct. Rössler and Duffing then provide useful comparisons.