Phase Diagram (State Space Plot) — lets-plot

A phase diagram (or state space plot) displays the trajectory of a dynamical system by plotting a variable against its derivative (x vs dx/dt). This visualization reveals the qualitative behavior of systems including fixed points, limit cycles, stability, and oscillation patterns. It is essential for analyzing differential equations without solving them explicitly.

Phase Diagram (State Space Plot) rendered with lets-plot

Python source (lets-plot)

""" anyplot.ai
phase-diagram: Phase Diagram (State Space Plot)
Library: letsplot 4.9.0 | Python 3.13.13
Quality: 90/100 | Updated: 2026-05-14
"""

import os

import numpy as np
import pandas as pd
from lets_plot import *


script_dir = os.path.dirname(os.path.abspath(__file__))
os.chdir(script_dir)

LetsPlot.setup_html()

# 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"

# Okabe-Ito colors
IMPRINT = ["#009E73", "#C475FD", "#4467A3"]

# Data - Damped harmonic oscillator with distinct parameters
# dx/dt = v, dv/dt = -omega^2 * x - gamma * v
np.random.seed(42)

omega = 1.5  # Natural frequency (different from bokeh's 2.0)
gamma = 0.5  # Higher damping coefficient (different from bokeh's 0.3)

# Three trajectories from different initial conditions
trajectories = []

initial_conditions = [
    (2.5, 1.0, "High position, rising"),
    (-2.0, -2.5, "Left position, falling"),
    (0.8, 3.0, "Near center, rising fast"),
]

for x0, v0, label in initial_conditions:
    t = np.linspace(0, 12, 400)
    dt = t[1] - t[0]

    x = np.zeros_like(t)
    v = np.zeros_like(t)
    x[0], v[0] = x0, v0

    # Euler integration for damped harmonic oscillator
    for i in range(1, len(t)):
        x[i] = x[i - 1] + v[i - 1] * dt
        v[i] = v[i - 1] + (-(omega**2) * x[i - 1] - gamma * v[i - 1]) * dt

    traj_df = pd.DataFrame({"x": x, "dx_dt": v, "time": t, "trajectory": label})
    trajectories.append(traj_df)

df = pd.concat(trajectories, ignore_index=True)

# Create phase diagram
anyplot_theme = theme(
    plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
    panel_background=element_rect(fill=PAGE_BG),
    panel_grid_major=element_line(color=INK_SOFT, size=0.3, linetype="solid"),
    axis_title=element_text(color=INK, size=20),
    axis_text=element_text(color=INK_SOFT, size=16),
    plot_title=element_text(color=INK, size=24),
    legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT),
    legend_text=element_text(color=INK_SOFT, size=14),
    legend_title=element_text(color=INK, size=16),
    legend_position="right",
)

plot = (
    ggplot(df, aes(x="x", y="dx_dt", color="trajectory"))
    + geom_path(size=1.3, alpha=0.85)
    + geom_point(
        mapping=aes(x="x", y="dx_dt"),
        data=df.groupby("trajectory").head(1),
        size=7,
        shape=21,
        fill=PAGE_BG,
        stroke=2.5,
        color=INK_SOFT,
    )
    # Mark the fixed point (equilibrium at origin)
    + geom_point(
        mapping=aes(x="x", y="dx_dt"),
        data=pd.DataFrame({"x": [0], "dx_dt": [0]}),
        color=INK_SOFT,
        size=10,
        shape=4,
        stroke=3,
        inherit_aes=False,
    )
    + scale_color_manual(values=IMPRINT)
    + labs(
        x="Position (x)", y="Velocity (dx/dt)", title="phase-diagram · letsplot · anyplot.ai", color="Initial Condition"
    )
    + theme_minimal()
    + anyplot_theme
    + ggsize(1600, 900)
)

# Save PNG (scale 3x for 4800 x 2700)
ggsave(plot, f"plot-{THEME}.png", path=".", scale=3)

# Save interactive HTML
ggsave(plot, f"plot-{THEME}.html", path=".")

Part of Phase Diagram (State Space Plot) on anyplot.ai.

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