Phase Diagram (State Space Plot) — Plotly

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 Plotly

Python source (Plotly)

""" anyplot.ai
phase-diagram: Phase Diagram (State Space Plot)
Library: plotly 6.7.0 | Python 3.13.13
Quality: 90/100 | Updated: 2026-05-14
"""

import os

import numpy as np
import plotly.graph_objects as go


# 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"
GRID = "rgba(26,26,23,0.20)" if THEME == "light" else "rgba(240,239,232,0.20)"
BRAND = "#009E73"  # Okabe-Ito position 1
ACCENT_1 = "#C475FD"  # Okabe-Ito position 2

# Data - Damped pendulum showing spiral convergence to equilibrium
np.random.seed(42)

# Parameters for damped harmonic oscillator: d²x/dt² + 2ζω₀(dx/dt) + ω₀²x = 0
omega_0 = 2.0
zeta = 0.15

# Time array
t = np.linspace(0, 15, 1500)
dt = t[1] - t[0]

# Damped frequency
omega_d = omega_0 * np.sqrt(1 - zeta**2)

# Trajectory 1: x(0) = 2.0, dx/dt(0) = 0
x0 = 2.0
v0 = 0.0
A = np.sqrt(x0**2 + ((v0 + zeta * omega_0 * x0) / omega_d) ** 2)
phi = np.arctan2((v0 + zeta * omega_0 * x0) / omega_d, x0)

x = A * np.exp(-zeta * omega_0 * t) * np.cos(omega_d * t - phi)
dx_dt = -zeta * omega_0 * A * np.exp(-zeta * omega_0 * t) * np.cos(omega_d * t - phi) - omega_d * A * np.exp(
    -zeta * omega_0 * t
) * np.sin(omega_d * t - phi)

# Trajectory 2: x(0) = -1.5, dx/dt(0) = 3.0
x0_2 = -1.5
v0_2 = 3.0
A2 = np.sqrt(x0_2**2 + ((v0_2 + zeta * omega_0 * x0_2) / omega_d) ** 2)
phi2 = np.arctan2((v0_2 + zeta * omega_0 * x0_2) / omega_d, x0_2)

x2 = A2 * np.exp(-zeta * omega_0 * t) * np.cos(omega_d * t - phi2)
dx_dt_2 = -zeta * omega_0 * A2 * np.exp(-zeta * omega_0 * t) * np.cos(omega_d * t - phi2) - omega_d * A2 * np.exp(
    -zeta * omega_0 * t
) * np.sin(omega_d * t - phi2)

# Create figure
fig = go.Figure()

# Main trajectory
fig.add_trace(
    go.Scatter(
        x=x,
        y=dx_dt,
        mode="lines+markers",
        name="Trajectory 1 (x₀=2.0, v₀=0)",
        line=dict(color=BRAND, width=3),
        marker=dict(size=7, color=BRAND, opacity=0.8),
        hovertemplate="x: %{x:.2f}<br>dx/dt: %{y:.2f}<extra></extra>",
    )
)

# Second trajectory
fig.add_trace(
    go.Scatter(
        x=x2,
        y=dx_dt_2,
        mode="lines+markers",
        name="Trajectory 2 (x₀=-1.5, v₀=3.0)",
        line=dict(color=ACCENT_1, width=3),
        marker=dict(size=7, color=ACCENT_1, opacity=0.8),
        hovertemplate="x: %{x:.2f}<br>dx/dt: %{y:.2f}<extra></extra>",
    )
)

# Fixed point (equilibrium at origin)
fig.add_trace(
    go.Scatter(
        x=[0],
        y=[0],
        mode="markers",
        name="Fixed Point (Stable)",
        marker=dict(size=18, color=INK, symbol="x", line=dict(width=3)),
        hovertemplate="Equilibrium<br>x=0, dx/dt=0<extra></extra>",
    )
)

# Initial conditions
fig.add_trace(
    go.Scatter(
        x=[x[0], x2[0]],
        y=[dx_dt[0], dx_dt_2[0]],
        mode="markers",
        name="Initial Conditions",
        marker=dict(size=14, color=INK_SOFT, symbol="circle"),
        hovertemplate="Initial: x=%{x:.2f}, dx/dt=%{y:.2f}<extra></extra>",
    )
)

# Direction arrows
arrow_indices = [200, 500, 900]
for idx in arrow_indices:
    # Arrow for trajectory 1
    fig.add_annotation(
        x=x[idx],
        y=dx_dt[idx],
        ax=x[idx - 30],
        ay=dx_dt[idx - 30],
        xref="x",
        yref="y",
        axref="x",
        ayref="y",
        showarrow=True,
        arrowhead=2,
        arrowsize=2,
        arrowwidth=2,
        arrowcolor=BRAND,
    )
    # Arrow for trajectory 2
    fig.add_annotation(
        x=x2[idx],
        y=dx_dt_2[idx],
        ax=x2[idx - 30],
        ay=dx_dt_2[idx - 30],
        xref="x",
        yref="y",
        axref="x",
        ayref="y",
        showarrow=True,
        arrowhead=2,
        arrowsize=2,
        arrowwidth=2,
        arrowcolor=ACCENT_1,
    )

# Update layout
fig.update_layout(
    title=dict(text="phase-diagram · plotly · anyplot.ai", font=dict(size=28, color=INK), x=0.5, xanchor="center"),
    xaxis=dict(
        title=dict(text="Position x", font=dict(size=22, color=INK)),
        tickfont=dict(size=18, color=INK_SOFT),
        gridcolor=GRID,
        gridwidth=1,
        zeroline=True,
        zerolinecolor=INK_SOFT,
        zerolinewidth=2,
        linecolor=INK_SOFT,
    ),
    yaxis=dict(
        title=dict(text="Velocity dx/dt", font=dict(size=22, color=INK)),
        tickfont=dict(size=18, color=INK_SOFT),
        gridcolor=GRID,
        gridwidth=1,
        zeroline=True,
        zerolinecolor=INK_SOFT,
        zerolinewidth=2,
        linecolor=INK_SOFT,
    ),
    legend=dict(
        font=dict(size=16, color=INK_SOFT), x=0.02, y=0.98, bgcolor=ELEVATED_BG, bordercolor=INK_SOFT, borderwidth=1
    ),
    paper_bgcolor=PAGE_BG,
    plot_bgcolor=PAGE_BG,
    font=dict(color=INK),
    margin=dict(l=80, r=40, t=100, b=80),
)

# Save outputs
fig.write_image(f"plot-{THEME}.png", width=1600, height=900, scale=3)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")

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

Phase Diagram (State Space Plot) — Plotly

Python source (Plotly)

Other implementations