A recurrence plot is a binary or distance-based matrix visualization that reveals when states in a dynamical system recur over time. Both axes represent time indices, and a point is plotted at position (i, j) when the system state at time i is sufficiently similar to the state at time j (distance below a threshold). The resulting symmetric matrix exposes hidden structure in complex, nonlinear dynamics — diagonal lines indicate determinism, vertical/horizontal lines reveal laminar states, and block structures signal regime changes.
""" anyplot.ai
recurrence-basic: Recurrence Plot for Nonlinear Time Series
Library: bokeh 3.9.1 | Python 3.13.13
Quality: 90/100 | Updated: 2026-06-10
"""
import io
import os
import sys
import time
from pathlib import Path
import numpy as np
from PIL import Image
# Workaround: remove current directory from import path to avoid shadowing
# the bokeh package with this file (bokeh.py) when run in-place
original_path = sys.path.copy()
sys.path = [p for p in sys.path if p != "" and not (os.path.isfile(os.path.join(p, "bokeh.py")) if p else False)]
try:
from bokeh.io import output_file, save
from bokeh.models import BasicTicker, ColorBar, ColumnDataSource, HoverTool, Label, LinearColorMapper
from bokeh.plotting import figure
from bokeh.resources import CDN
from selenium import webdriver
from selenium.webdriver.chrome.options import Options
finally:
sys.path = original_path
# Theme tokens
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 sequential colormap: brand green → blue (256 stops, single-polarity)
_c0 = np.array([0x00, 0x9E, 0x73]) # #009E73
_c1 = np.array([0x44, 0x67, 0xA3]) # #4467A3
IMPRINT_SEQ256 = [
"#{:02X}{:02X}{:02X}".format(*(int(round(v)) for v in (_c0 + (_c1 - _c0) * t / 255.0))) for t in range(256)
]
# Data — Lorenz attractor x-component
np.random.seed(42)
dt = 0.01
num_steps = 5000
lx, ly, lz = 1.0, 1.0, 1.0
sigma, rho, beta = 10.0, 28.0, 8.0 / 3.0
trajectory = np.empty(num_steps)
for step in range(num_steps):
dx = sigma * (ly - lx) * dt
dy = (lx * (rho - lz) - ly) * dt
dz = (lx * ly - beta * lz) * dt
lx, ly, lz = lx + dx, ly + dy, lz + dz
trajectory[step] = lx
signal = trajectory[::10]
n_points = len(signal)
# Time-delay embedding (Takens' theorem): dim=3, delay=5
tau = 5
dim = 3
n_embedded = n_points - (dim - 1) * tau
embedded = np.empty((n_embedded, dim))
for d in range(dim):
embedded[:, d] = signal[d * tau : d * tau + n_embedded]
# Pairwise Euclidean distance matrix
diff = embedded[:, np.newaxis, :] - embedded[np.newaxis, :, :]
dist_matrix = np.sqrt(np.sum(diff**2, axis=2))
threshold = np.percentile(dist_matrix, 10)
max_dist = dist_matrix.max()
dist_normalized = dist_matrix / max_dist
dist_flipped = dist_normalized[::-1, :] # row 0 at top
color_mapper = LinearColorMapper(palette=IMPRINT_SEQ256, low=0.0, high=1.0)
# Plot — square 2400×2400 canonical
n = n_embedded
p = figure(
width=2400,
height=2400,
title="recurrence-basic · python · bokeh · anyplot.ai",
x_axis_label="Time Index (Lorenz x-component, dt=0.01)",
y_axis_label="Time Index (Lorenz x-component, dt=0.01)",
toolbar_location=None,
tools="",
x_range=(0, n),
y_range=(0, n),
min_border_bottom=160,
min_border_left=180,
min_border_top=110,
min_border_right=50,
)
p.image(image=[dist_flipped], x=0, y=0, dw=n, dh=n, color_mapper=color_mapper)
color_bar = ColorBar(
color_mapper=color_mapper,
ticker=BasicTicker(desired_num_ticks=6),
label_standoff=16,
border_line_color=None,
background_fill_color=PAGE_BG,
location=(0, 0),
title="Normalized Distance",
title_text_font_size="28pt",
title_text_color=INK,
major_label_text_font_size="24pt",
major_label_text_color=INK_SOFT,
width=40,
padding=30,
)
p.add_layout(color_bar, "right")
p.add_layout(
Label(
x=200,
y=260,
text="Laminar regime",
text_font_size="28pt",
text_color=INK,
text_font_style="bold",
background_fill_color=ELEVATED_BG,
background_fill_alpha=0.88,
)
)
p.add_layout(
Label(
x=120,
y=160,
text="Deterministic diagonals",
text_font_size="28pt",
text_color=INK,
text_font_style="bold",
background_fill_color=ELEVATED_BG,
background_fill_alpha=0.88,
angle=0.78,
)
)
p.add_layout(
Label(
x=10,
y=n - 25,
text=f"Recurrence threshold ε = {threshold:.1f} (10th percentile)",
text_font_size="24pt",
text_color=INK,
background_fill_color=ELEVATED_BG,
background_fill_alpha=0.88,
)
)
# HoverTool via invisible scatter overlay (idiomatic Bokeh for image plots)
hover_step = 20
hover_xs, hover_ys, hover_dists, hover_recs = [], [], [], []
for i in range(0, n, hover_step):
for j in range(0, n, hover_step):
hover_xs.append(i + hover_step // 2)
hover_ys.append(j + hover_step // 2)
dist_ij = dist_matrix[i, j]
hover_dists.append(round(float(dist_ij), 2))
hover_recs.append("Yes" if dist_ij <= threshold else "No")
hover_source = ColumnDataSource(data={"x": hover_xs, "y": hover_ys, "distance": hover_dists, "recurrent": hover_recs})
invisible_scatter = p.scatter(x="x", y="y", source=hover_source, size=hover_step, fill_alpha=0, line_alpha=0)
p.add_tools(
HoverTool(
renderers=[invisible_scatter],
tooltips=[
("Time i", "@x"),
("Time j", "@y"),
("Distance", "@distance"),
("Recurrent (d < {:.1f})".format(threshold), "@recurrent"),
],
)
)
# Style — theme-adaptive chrome
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.axis_label_text_color = INK
p.yaxis.axis_label_text_color = INK
p.xaxis.major_label_text_font_size = "34pt"
p.yaxis.major_label_text_font_size = "34pt"
p.xaxis.major_label_text_color = INK_SOFT
p.yaxis.major_label_text_color = INK_SOFT
p.xgrid.grid_line_color = None
p.ygrid.grid_line_color = None
p.axis.minor_tick_line_color = None
p.xaxis.axis_line_color = INK_SOFT
p.yaxis.axis_line_color = INK_SOFT
p.xaxis.major_tick_line_color = INK_SOFT
p.yaxis.major_tick_line_color = INK_SOFT
p.outline_line_color = INK_SOFT
p.background_fill_color = PAGE_BG
p.border_fill_color = PAGE_BG
# Save HTML then screenshot with headless Chrome (export_png not used — snap shim fails)
output_file(f"plot-{THEME}.html")
save(p, resources=CDN)
# Window is H+200 tall so the full bokeh canvas renders; PIL crops to W×H.
W, H = 2400, 2400
opts = Options()
for arg in (
"--headless=new",
"--no-sandbox",
"--disable-dev-shm-usage",
"--disable-gpu",
f"--window-size={W},{H + 200}",
"--hide-scrollbars",
"--force-device-scale-factor=1",
):
opts.add_argument(arg)
driver = webdriver.Chrome(options=opts)
driver.set_window_size(W, H + 200)
driver.get(f"file://{Path(f'plot-{THEME}.html').resolve()}")
time.sleep(3)
raw = driver.get_screenshot_as_png()
driver.quit()
Image.open(io.BytesIO(raw)).crop((0, 0, W, H)).save(f"plot-{THEME}.png")
Part of Recurrence Plot for Nonlinear Time Series on anyplot.ai.