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.
""" anyplot.ai
bifurcation-basic: Bifurcation Diagram for Dynamical Systems
Library: altair 6.2.1 | Python 3.13.14
Quality: 90/100 | Updated: 2026-06-17
"""
import os
import altair as alt
import numpy as np
import pandas as pd
from PIL import Image
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
# Imprint palette — semantic mapping: stable→green, period-doubling→amber(caution), chaos→red
STABLE_COLOR = "#009E73" # Imprint position 1 — brand green = stable / ok
PERIOD_COLOR = "#DDCC77" # Imprint amber anchor — warning / transition
CHAOS_COLOR = "#AE3030" # Imprint position 5 — matte red = bad / chaos
# Data — Logistic map: x(n+1) = r * x(n) * (1 - x(n))
np.random.seed(42)
r_values = np.linspace(2.5, 4.0, 2000)
transient = 200
iterations = 100
r_all = []
x_all = []
for r in r_values:
x = 0.5
for _ in range(transient):
x = r * x * (1.0 - x)
for _ in range(iterations):
x = r * x * (1.0 - x)
r_all.append(r)
x_all.append(x)
df = pd.DataFrame({"r": r_all, "x": x_all})
# Classify regions for semantic coloring
df["region"] = np.where(df["r"] < 3.0, "Stable", np.where(df["r"] < 3.57, "Period-doubling", "Chaotic"))
# Key bifurcation points — staggered y to avoid label overlap near the chaos onset
bifurcation_points = pd.DataFrame(
{
"r": [3.0, 3.449, 3.544, 3.5699],
"label": ["Period 2 (r≈3.0)", "Period 4 (r≈3.45)", "Period 8", "Chaos onset"],
"y": [0.90, 0.90, 0.55, 0.82],
}
)
# Selection for interactive crosshair on nearest point
nearest = alt.selection_point(nearest=True, on="pointerover", fields=["r"], empty=False)
# Semantic color scale from Imprint palette
region_color = alt.Scale(
domain=["Stable", "Period-doubling", "Chaotic"], range=[STABLE_COLOR, PERIOD_COLOR, CHAOS_COLOR]
)
# Base scatter layer — density-adapted size and opacity for 200K points
points = (
alt.Chart(df)
.mark_circle(size=1.5, opacity=0.18)
.encode(
x=alt.X(
"r:Q",
title="Growth Rate (r)",
scale=alt.Scale(domain=[2.5, 4.0], nice=False),
axis=alt.Axis(tickCount=7, titleColor=INK, labelColor=INK_SOFT, domain=False),
),
y=alt.Y(
"x:Q",
title="Steady-State Population (x)",
scale=alt.Scale(domain=[0, 1.0], nice=False),
axis=alt.Axis(tickCount=6, titleColor=INK, labelColor=INK_SOFT, domain=False),
),
color=alt.Color(
"region:N",
scale=region_color,
legend=alt.Legend(
title="Regime",
titleFontSize=10,
labelFontSize=10,
orient="top-right",
offset=-10,
fillColor=ELEVATED_BG,
strokeColor=INK_SOFT,
padding=8,
cornerRadius=4,
labelColor=INK_SOFT,
titleColor=INK,
),
),
tooltip=[alt.Tooltip("r:Q", title="r", format=".4f"), alt.Tooltip("x:Q", title="x", format=".4f"), "region:N"],
)
)
# Transparent layer for nearest-point selection (voronoi-like hover)
voronoi = (
alt.Chart(df.sample(n=5000, random_state=42))
.mark_point(size=1, opacity=0)
.encode(x="r:Q", y="x:Q")
.add_params(nearest)
)
# Vertical crosshair following pointer
crosshair = (
alt.Chart(df.sample(n=5000, random_state=42))
.mark_rule(color=INK_SOFT, strokeWidth=1.5, opacity=0.5)
.encode(x="r:Q")
.transform_filter(nearest)
)
# Dashed vertical rules at bifurcation points
rules = (
alt.Chart(bifurcation_points)
.mark_rule(strokeDash=[5, 4], strokeWidth=1.2, opacity=0.30, color=INK_MUTED)
.encode(x="r:Q")
)
# Rotated text labels at bifurcation points — fontSize=12 for clear readability at full res
labels = (
alt.Chart(bifurcation_points)
.mark_text(fontSize=12, fontWeight="bold", color=INK_SOFT, angle=270, align="left", dx=0, dy=-8)
.encode(x="r:Q", y="y:Q", text="label:N")
)
# Compose layers with theme-adaptive chrome
chart = (
(points + voronoi + crosshair + rules + labels)
.properties(
width=620,
height=275,
background=PAGE_BG,
title=alt.Title(
"bifurcation-basic · python · altair · anyplot.ai",
fontSize=16,
fontWeight="bold",
color=INK,
subtitle="Logistic map period-doubling cascade from stability to chaos",
subtitleFontSize=11,
subtitleColor=INK_SOFT,
subtitlePadding=4,
),
)
.configure_view(fill=PAGE_BG, strokeWidth=0)
.configure_axis(
labelFontSize=10,
titleFontSize=12,
titlePadding=10,
grid=True,
gridOpacity=0.15,
gridColor=INK,
domainColor=INK_SOFT,
tickColor=INK_SOFT,
)
.configure_title(color=INK)
.configure_legend(
fillColor=ELEVATED_BG,
strokeColor=INK_SOFT,
labelColor=INK_SOFT,
titleColor=INK,
labelFontSize=10,
titleFontSize=10,
)
)
# Save PNG then pad to exactly 3200×1800 (landscape target)
TW, TH = 3200, 1800
chart.save(f"plot-{THEME}.png", scale_factor=4.0)
_img = Image.open(f"plot-{THEME}.png").convert("RGB")
_w, _h = _img.size
if _w > TW or _h > TH:
raise SystemExit(
f"altair vl-convert produced {_w}×{_h}, exceeds target {TW}×{TH}. "
"Shrink chart .properties(width=, height=) values and re-render."
)
if _w < TW or _h < TH:
_canvas = Image.new("RGB", (TW, TH), PAGE_BG)
_canvas.paste(_img, ((TW - _w) // 2, (TH - _h) // 2))
_canvas.save(f"plot-{THEME}.png")
# Save interactive HTML
chart.save(f"plot-{THEME}.html")
Part of Bifurcation Diagram for Dynamical Systems on anyplot.ai.