Phase Diagram (State Space Plot) — Seaborn

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 Seaborn

Python source (Seaborn)

""" anyplot.ai
phase-diagram: Phase Diagram (State Space Plot)
Library: seaborn 0.13.2 | Python 3.13.13
Quality: 93/100 | Updated: 2026-05-14
"""

import os

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns


# 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 palette (first series always #009E73)
IMPRINT = ["#009E73", "#C475FD", "#4467A3", "#BD8233"]

# Data: Damped harmonic oscillator phase trajectories
np.random.seed(42)

# System parameters for damped oscillator: d²x/dt² + 2*zeta*omega*dx/dt + omega²*x = 0
omega = 2 * np.pi
zeta = 0.15

# Generate multiple trajectories from different initial conditions
t = np.linspace(0, 5, 500)
trajectories = []
initial_conditions = [(2.0, 0.0), (0.0, 8.0), (-1.5, -5.0), (1.0, 4.0)]

for x0, v0 in initial_conditions:
    # Analytical solution for underdamped oscillator
    omega_d = omega * np.sqrt(1 - zeta**2)
    A = np.sqrt(x0**2 + ((zeta * omega * x0 + v0) / omega_d) ** 2)
    phi = np.arctan2(omega_d * x0, zeta * omega * x0 + v0)

    # Position and velocity (derivative)
    x = A * np.exp(-zeta * omega * t) * np.sin(omega_d * t + phi)
    dx_dt = (
        A
        * np.exp(-zeta * omega * t)
        * (-zeta * omega * np.sin(omega_d * t + phi) + omega_d * np.cos(omega_d * t + phi))
    )

    trajectories.append((x, dx_dt, f"({x0}, {v0})"))

# Create DataFrame for seaborn
data = []
for x, dx_dt, label in trajectories:
    for i in range(len(x)):
        data.append({"Position (x)": x[i], "Velocity (dx/dt)": dx_dt[i], "Initial Condition": label, "Time": t[i]})
df = pd.DataFrame(data)

# Configure seaborn theme
sns.set_theme(
    style="ticks",
    rc={
        "figure.facecolor": PAGE_BG,
        "axes.facecolor": PAGE_BG,
        "axes.edgecolor": INK_SOFT,
        "axes.labelcolor": INK,
        "text.color": INK,
        "xtick.color": INK_SOFT,
        "ytick.color": INK_SOFT,
        "grid.color": INK,
        "grid.alpha": 0.10,
        "legend.facecolor": ELEVATED_BG,
        "legend.edgecolor": INK_SOFT,
    },
)

# Plot
fig, ax = plt.subplots(figsize=(16, 9))

# Plot trajectories with gradient alpha to reduce density
sns.lineplot(
    data=df,
    x="Position (x)",
    y="Velocity (dx/dt)",
    hue="Initial Condition",
    palette=IMPRINT,
    linewidth=3,
    alpha=0.8,
    legend=True,
    ax=ax,
    sort=False,
)

# Add starting points as larger markers
for i, (x, dx_dt, _label) in enumerate(trajectories):
    ax.scatter(x[0], dx_dt[0], s=300, color=IMPRINT[i], zorder=5, edgecolor=PAGE_BG, linewidth=2)

# Add fixed point (equilibrium at origin)
ax.scatter(0, 0, s=400, color=INK, marker="x", linewidth=3.5, zorder=6, label="Equilibrium")

# Add direction arrows on trajectories
for i, (x, dx_dt, _label) in enumerate(trajectories):
    arrow_indices = [50, 150, 300]
    for idx in arrow_indices:
        if idx < len(x) - 1:
            dx = x[idx + 1] - x[idx]
            dy = dx_dt[idx + 1] - dx_dt[idx]
            ax.annotate(
                "",
                xy=(x[idx] + dx * 0.5, dx_dt[idx] + dy * 0.5),
                xytext=(x[idx], dx_dt[idx]),
                arrowprops={"arrowstyle": "->", "color": IMPRINT[i], "lw": 2},
            )

# Styling
ax.set_xlabel("Position (x)", fontsize=20, color=INK)
ax.set_ylabel("Velocity (dx/dt)", fontsize=20, color=INK)
ax.set_title("phase-diagram · seaborn · anyplot.ai", fontsize=24, color=INK)
ax.tick_params(axis="both", labelsize=16, colors=INK_SOFT)

# Remove top and right spines
ax.spines["top"].set_visible(False)
ax.spines["right"].set_visible(False)
for s in ("left", "bottom"):
    ax.spines[s].set_color(INK_SOFT)

# Add zero lines for reference
ax.axhline(y=0, color=INK_SOFT, linestyle="--", linewidth=1.5, alpha=0.4)
ax.axvline(x=0, color=INK_SOFT, linestyle="--", linewidth=1.5, alpha=0.4)

# Adjust legend
legend = ax.legend(fontsize=14, loc="upper right", title="Initial Condition", title_fontsize=16)
legend.get_frame().set_facecolor(ELEVATED_BG)
legend.get_frame().set_edgecolor(INK_SOFT)

plt.tight_layout()
plt.savefig(f"plot-{THEME}.png", dpi=300, bbox_inches="tight", facecolor=PAGE_BG)

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

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