Bifurcation Diagram for Dynamical Systems — Pygal

A bifurcation diagram shows how the steady-state behavior of a dynamical system changes as a control parameter varies. By plotting the long-term values of a state variable against a continuously varied parameter, it reveals transitions from stable fixed points through period-doubling cascades to chaotic regimes. The classic example is the logistic map, where the route to chaos is clearly visible as the growth rate parameter increases.

Bifurcation Diagram for Dynamical Systems rendered with Pygal

Python source (Pygal)

""" anyplot.ai
bifurcation-basic: Bifurcation Diagram for Dynamical Systems
Library: pygal 3.1.0 | Python 3.13.13
Quality: 82/100 | Updated: 2026-06-17
"""

import os

import numpy as np
import pygal
from pygal.style import Style


# Theme tokens — Imprint palette chrome
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"

# Imprint categorical palette — 8 hues, hybrid-v3 sort
IMPRINT_PALETTE = ("#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477", "#99B314")

# Logistic map data — x(n+1) = r * x(n) * (1 - x(n))
np.random.seed(42)
TRANSIENT = 200
ITERATIONS = 100
x0 = 0.1 + np.random.uniform(-0.01, 0.01)

# Key bifurcation thresholds
R_PERIOD2 = 3.0
R_PERIOD4 = 3.449
R_PERIOD8 = 3.544
R_CHAOS = 3.57

# Variable-density sampling: more points where structure is richer
r_stable = np.linspace(2.5, R_PERIOD2, 250)
r_periodic = np.linspace(R_PERIOD2, R_CHAOS, 500)
r_chaotic = np.linspace(R_CHAOS, 4.0, 700)
r_values = np.concatenate([r_stable, r_periodic, r_chaotic])

# Three dynamical regions mapped to first three Imprint palette positions
regions = {
    "Stable Fixed Point": (2.5, R_PERIOD2, IMPRINT_PALETTE[0]),
    "Period-Doubling Cascade": (R_PERIOD2, R_CHAOS, IMPRINT_PALETTE[1]),
    "Chaotic Regime": (R_CHAOS, 4.0, IMPRINT_PALETTE[2]),
}

region_data = {name: [] for name in regions}

for r in r_values:
    x = x0
    for _ in range(TRANSIENT):
        x = r * x * (1.0 - x)
    for _ in range(ITERATIONS):
        x = r * x * (1.0 - x)
        for name, (lo, hi, _) in regions.items():
            if lo <= r < hi or (name == "Chaotic Regime" and r == 4.0):
                region_data[name].append(
                    {"value": (round(float(r), 5), round(float(x), 5)), "label": f"r={r:.4f}, x={x:.4f}"}
                )
                break

# Downsample each region to balance visual density
max_per_region = {"Stable Fixed Point": 6000, "Period-Doubling Cascade": 18000, "Chaotic Regime": 28000}
for name in region_data:
    pts = region_data[name]
    cap = max_per_region[name]
    if len(pts) > cap:
        idx = np.random.choice(len(pts), cap, replace=False)
        idx.sort()
        region_data[name] = [pts[i] for i in idx]

# Color tuple: 3 Imprint data series + INK_MUTED for dashed annotation lines
region_colors = tuple(color for _, (_, _, color) in regions.items())
all_colors = region_colors + (INK_MUTED,)

custom_style = Style(
    background=PAGE_BG,
    plot_background=PAGE_BG,
    foreground=INK,
    foreground_strong=INK,
    foreground_subtle=INK_MUTED,
    guide_stroke_color=INK_MUTED,
    guide_stroke_dasharray="3, 8",
    colors=all_colors,
    title_font_size=66,
    label_font_size=56,
    major_label_font_size=44,
    legend_font_size=44,
    value_font_size=36,
    tooltip_font_size=32,
    stroke_width=2.5,
    opacity=0.55,
    opacity_hover=1.0,
)

chart = pygal.XY(
    width=3200,
    height=1800,
    style=custom_style,
    title="bifurcation-basic · python · pygal · anyplot.ai",
    x_title="Growth Rate Parameter (r)",
    y_title="Steady-State Population (xₙ)",
    show_legend=True,
    legend_at_bottom=True,
    legend_at_bottom_columns=2,
    legend_box_size=22,
    stroke=False,
    dots_size=1.2,
    show_x_guides=True,
    show_y_guides=True,
    x_value_formatter=lambda v: f"{v:.3f}",
    value_formatter=lambda v: f"{v:.4f}",
    margin_bottom=160,
    margin_left=70,
    margin_right=50,
    margin_top=55,
    xrange=(2.5, 4.0),
    range=(0.0, 1.0),
    print_values=False,
    print_zeroes=False,
    js=[],
    x_labels=[2.5, R_PERIOD2, 3.2, R_PERIOD4, R_PERIOD8, 3.7, 3.8, 4.0],
    x_labels_major=[R_PERIOD2, R_PERIOD4, R_PERIOD8],
    y_labels=[0.0, 0.2, 0.4, 0.6, 0.8, 1.0],
    truncate_legend=-1,
    no_data_text="",
    show_x_labels=True,
    show_y_labels=True,
    allow_interruptions=True,
    show_minor_x_labels=True,
    spacing=25,
    include_x_axis=True,
)

# Add each dynamical region as a separate series with per-point tooltip metadata
for name in regions:
    lo, hi, _ = regions[name]
    chart.add(
        f"{name} (r≈{lo:.1f}–{hi:.2f})", region_data[name], stroke=False, show_dots=True, allow_interruptions=True
    )

# Dashed vertical lines at key bifurcation thresholds — no secondary axis
annotation_points = [
    (R_PERIOD2, "r≈3.0: Period-2 onset"),
    (R_PERIOD4, "r≈3.449: Period-4 onset"),
    (R_PERIOD8, "r≈3.544: Period-8 onset"),
]

annotation_data = []
for r_val, label in annotation_points:
    annotation_data.append({"value": (r_val, 0.0), "label": label})
    annotation_data.append({"value": (r_val, 1.0), "label": label})
    annotation_data.append(None)

chart.add(
    "Bifurcation Points",
    annotation_data,
    stroke=True,
    stroke_style={"width": 2.5, "dasharray": "10, 5"},
    show_dots=False,
    dots_size=0,
)

chart.render_to_png(f"plot-{THEME}.png")
with open(f"plot-{THEME}.html", "wb") as f:
    f.write(chart.render())

Part of Bifurcation Diagram for Dynamical Systems on anyplot.ai.

Bifurcation Diagram for Dynamical Systems — Pygal

Python source (Pygal)

Other implementations