Phase Diagram (State Space Plot) — plotnine

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 plotnine

Python source (plotnine)

""" anyplot.ai
phase-diagram: Phase Diagram (State Space Plot)
Library: plotnine 0.15.4 | Python 3.13.13
Quality: 92/100 | Updated: 2026-05-14
"""

import os

import numpy as np
import pandas as pd
from plotnine import (
    aes,
    element_line,
    element_rect,
    element_text,
    geom_hline,
    geom_path,
    geom_point,
    geom_vline,
    ggplot,
    ggsave,
    labs,
    scale_color_cmap,
    theme,
    theme_minimal,
)


# 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"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
BRAND = "#009E73"  # Okabe-Ito position 1

# Data: Damped harmonic oscillator (spiral trajectory converging to equilibrium)
np.random.seed(42)

# System parameters
omega = 2 * np.pi  # Natural frequency
gamma = 0.15  # Damping coefficient

# Time array for smooth trajectory
t = np.linspace(0, 8, 800)

# Solution for damped harmonic oscillator: x = A * exp(-gamma*t) * cos(omega*t)
A = 2.0  # Initial amplitude
x = A * np.exp(-gamma * t) * np.cos(omega * t)
dx_dt = A * np.exp(-gamma * t) * (-gamma * np.cos(omega * t) - omega * np.sin(omega * t))

# Create DataFrame with time for color gradient
df = pd.DataFrame({"x": x, "dx_dt": dx_dt, "t": t})

# Theme-adaptive chrome
anyplot_theme = theme(
    plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
    panel_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
    panel_grid_major=element_line(color=INK, size=0.3, alpha=0.10),
    panel_grid_minor=element_line(color=INK, size=0.2, alpha=0.05),
    panel_border=element_rect(color=INK_SOFT, fill=None, size=0.6),
    axis_title=element_text(color=INK, size=20),
    axis_text=element_text(color=INK_SOFT, size=16),
    axis_line=element_line(color=INK_SOFT, size=0.5),
    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=16),
    legend_title=element_text(color=INK, size=16),
    figure_size=(16, 9),
)

# Create phase diagram
plot = (
    ggplot(df, aes(x="x", y="dx_dt", color="t"))
    + geom_path(size=1.5, alpha=0.9)
    + geom_point(data=df.iloc[[0]], size=5, color=BRAND, show_legend=False)  # Start point
    + geom_point(data=df.iloc[[-1]], size=5, color=INK_SOFT, shape="s", show_legend=False)  # End point
    + geom_hline(yintercept=0, linetype="dashed", color=INK_MUTED, alpha=0.5, size=0.5)
    + geom_vline(xintercept=0, linetype="dashed", color=INK_MUTED, alpha=0.5, size=0.5)
    + scale_color_cmap(cmap_name="viridis", name="Time (s)")
    + labs(x="Position x", y="Velocity dx/dt", title="phase-diagram · plotnine · anyplot.ai")
    + theme_minimal()
    + anyplot_theme
)

# Save
ggsave(plot, f"plot-{THEME}.png", dpi=300, verbose=False)

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

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