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: bokeh 3.9.1 | Python 3.13.13
Quality: 93/100 | Updated: 2026-06-17
"""
import os
import time
from pathlib import Path
import numpy as np
from bokeh.io import output_file, save
from bokeh.models import HoverTool, Label, Range1d, Span
from bokeh.plotting import ColumnDataSource, figure
from selenium import webdriver
from selenium.webdriver.chrome.options import Options
# Theme-adaptive chrome
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"
# Imprint palette — regime encoding tells the route-to-chaos story.
# Semantic ordering: stable=brand green, intermediate periods cool blue/lavender,
# chaos=matte-red (Imprint semantic anchor for instability). First series #009E73.
IMPRINT_STABLE = "#009E73" # position 1 — stable fixed point (good/safe → green)
IMPRINT_PERIOD2 = "#4467A3" # position 3 — period-2 band
IMPRINT_PERIOD4 = "#C475FD" # position 2 — period-4 / higher period
IMPRINT_CHAOS = "#AE3030" # position 5 — chaotic regime (instability → red anchor)
# Data — logistic map: x(n+1) = r * x(n) * (1 - x(n))
r_min, r_max = 2.5, 4.0
n_r = 3000
n_transient = 300
n_keep = 150
r_values = np.linspace(r_min, r_max, n_r)
all_r = np.repeat(r_values, n_keep)
all_x = np.empty_like(all_r)
idx = 0
for r in r_values:
x = 0.5
for _ in range(n_transient):
x = r * x * (1.0 - x)
for _ in range(n_keep):
x = r * x * (1.0 - x)
all_x[idx] = x
idx += 1
# Vectorized regime-based color + density-aware alpha (np.select, no Python loop).
# Chaos alpha lifted from the old 0.15 to 0.30 so the dense red fractal holds
# contrast on both the cream and near-black surfaces.
r_cuts = [all_r < 3.0, all_r < 3.449, all_r < 3.5699]
colors = np.select(r_cuts, [IMPRINT_STABLE, IMPRINT_PERIOD2, IMPRINT_PERIOD4], default=IMPRINT_CHAOS)
alphas = np.select(r_cuts, [0.65, 0.50, 0.40], default=0.30)
source = ColumnDataSource(data={"r": all_r, "x": all_x, "color": colors, "alpha": alphas})
# Plot — canonical 3200×1800 landscape canvas
title = "bifurcation-basic · python · bokeh · anyplot.ai"
p = figure(
width=3200,
height=1800,
title=title,
x_axis_label="Growth Rate (r)",
y_axis_label="Steady-State Population (x)",
x_range=Range1d(r_min - 0.02, r_max + 0.02),
y_range=Range1d(-0.04, 1.04),
toolbar_location=None,
min_border_bottom=160,
min_border_left=180,
min_border_top=110,
min_border_right=60,
tools="pan,wheel_zoom,box_zoom,reset,save",
active_scroll="wheel_zoom",
)
scatter = p.scatter(x="r", y="x", source=source, size=2, color="color", alpha="alpha", line_color=None)
# HoverTool — Bokeh-distinctive interactive feature (live in the HTML artifact)
hover = HoverTool(
renderers=[scatter], tooltips=[("r", "@r{0.000}"), ("x", "@x{0.0000}")], point_policy="snap_to_data", mode="mouse"
)
p.add_tools(hover)
# Vertical guides at the key bifurcation points
bif_points = [3.0, 3.449, 3.5699]
for r_bif in bif_points:
p.add_layout(
Span(location=r_bif, dimension="height", line_color=INK_SOFT, line_width=2, line_alpha=0.35, line_dash="dashed")
)
# Annotations — boxed so they read over the dense chaotic fan; the chaos label
# now sits high inside the structure it describes instead of floating at y≈0.05.
annotations = [
(3.0, 0.70, "r ≈ 3.0\nPeriod-2", "right"),
(3.449, 0.90, "r ≈ 3.449\nPeriod-4", "right"),
(3.5699, 0.97, "r ≈ 3.57\nOnset of chaos", "left"),
]
for r_bif, y_pos, label_text, align in annotations:
p.add_layout(
Label(
x=r_bif,
y=y_pos,
text=label_text,
text_font_size="30pt",
text_font_style="bold",
text_color=INK,
text_align=align,
x_offset=14 if align == "left" else -14,
background_fill_color=ELEVATED_BG,
background_fill_alpha=0.82,
border_line_color=INK_SOFT,
border_line_alpha=0.4,
)
)
# Style — typography
p.title.text_font_size = "50pt"
p.title.text_color = INK
p.xaxis.axis_label_text_font_size = "42pt"
p.yaxis.axis_label_text_font_size = "42pt"
p.xaxis.major_label_text_font_size = "34pt"
p.yaxis.major_label_text_font_size = "34pt"
p.xaxis.axis_label_text_color = INK
p.yaxis.axis_label_text_color = INK
p.xaxis.major_label_text_color = INK_SOFT
p.yaxis.major_label_text_color = INK_SOFT
# Chrome — clean L-frame, ticks/axis lines removed for a minimal scatter
p.xaxis.axis_line_color = None
p.yaxis.axis_line_color = None
p.xaxis.major_tick_line_color = None
p.yaxis.major_tick_line_color = None
p.xaxis.minor_tick_line_color = None
p.yaxis.minor_tick_line_color = None
p.xgrid.grid_line_color = INK
p.ygrid.grid_line_color = INK
p.xgrid.grid_line_alpha = 0.12
p.ygrid.grid_line_alpha = 0.12
p.background_fill_color = PAGE_BG
p.border_fill_color = PAGE_BG
p.outline_line_color = None
p.xaxis.ticker.desired_num_ticks = 12
p.yaxis.ticker.desired_num_ticks = 8
# Save — interactive HTML artifact + headless-Chrome screenshot at the exact canvas size
output_file(f"plot-{THEME}.html", title="Bifurcation Diagram")
save(p)
W, H = 3200, 1800
opts = Options()
for arg in (
"--headless=new",
"--no-sandbox",
"--disable-dev-shm-usage",
"--disable-gpu",
f"--window-size={W},{H}",
"--hide-scrollbars",
):
opts.add_argument(arg)
driver = webdriver.Chrome(options=opts)
driver.set_window_size(W, H)
driver.get(f"file://{Path(f'plot-{THEME}.html').resolve()}")
time.sleep(3) # let bokeh's JS render the canvas
# Headless Chrome reserves a strip of window chrome, so the inner viewport (what
# save_screenshot captures) lands ~140px short of the requested window height.
# Measure the delta and resize so the viewport is EXACTLY 3200×1800.
inner_w, inner_h = driver.execute_script("return [window.innerWidth, window.innerHeight];")
driver.set_window_size(W + (W - inner_w), H + (H - inner_h))
time.sleep(1)
driver.save_screenshot(f"plot-{THEME}.png")
driver.quit()
Part of Bifurcation Diagram for Dynamical Systems on anyplot.ai.