Lorenz attractor

Python Code

"""Dynamical Systems — Lorenz attractor (σ=10, ρ=28, β=8/3)."""


import numpy as np
import plotly.graph_objects as go
from scipy.integrate import solve_ivp


SIGMA, RHO, BETA = 10.0, 28.0, 8.0 / 3.0

# Full-spectrum rainbow, swept left → right across the attractor.
RAINBOW = [
    [0.00, "#ff1f3d"],   # red
    [0.18, "#ff7a18"],   # orange
    [0.36, "#ffe000"],   # yellow
    [0.58, "#3ddc5b"],   # green
    [0.78, "#1fc8e0"],   # cyan
    [1.00, "#2b6bff"],   # blue
]


def lorenz(t, state):
    x, y, z = state
    return [
        SIGMA * (y - x),
        x * (RHO - z) - y,
        x * y - BETA * z,
    ]


def generate():
    print("integrating Lorenz attractor …")

    # One long orbit — the strange attractor is dense enough that a single
    # trajectory, drawn as a haze of translucent points, fills out the set.
    ic = [0.1, 0.0, 0.0]
    t_span = (0, 130)
    t_eval = np.linspace(0, 130, 55000)

    sol = solve_ivp(lorenz, t_span, ic, t_eval=t_eval, rtol=1e-9, atol=1e-11)
    x, y, z = np.round(sol.y, 2)

    print(f"  integrated {len(x)} points, x range [{x.min():.1f}, {x.max():.1f}]")

    fig = go.Figure()

    # Classic (x, z) butterfly silhouette as a glowing point cloud: tiny
    # semi-transparent markers accumulate into smooth luminous ribbons, and the
    # spectrum sweeps across x so the wings run red → blue.
    fig.add_trace(go.Scattergl(
        x=x, y=z,
        mode="markers",
        marker=dict(
            size=2.8,
            color=x,
            colorscale=RAINBOW,
            opacity=0.55,
            showscale=False,
            line=dict(width=0),
        ),
        hoverinfo="skip",
        name="Lorenz attractor",
    ))

    axis = dict(visible=False, showgrid=False, zeroline=False,
                showticklabels=False)
    fig.update_layout(
        xaxis=axis,
        yaxis=dict(visible=False, showgrid=False, zeroline=False,
                   showticklabels=False, scaleanchor="x", scaleratio=1),
        paper_bgcolor="#000000",
        plot_bgcolor="#000000",
        showlegend=False,
        margin=dict(t=0, b=0, l=0, r=0),
        height=800,
    )

    return fig


if __name__ == "__main__":
    generate()

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