feat(pygal): implement phase-diagram (#3061)

## Implementation: `phase-diagram` - pygal

Implements the **pygal** version of `phase-diagram`.

**File:** `plots/phase-diagram/implementations/pygal.py`

**Parent Issue:** #3004

---
:robot: *[impl-generate
workflow](https://github.com/MarkusNeusinger/pyplots/actions/runs/20617709332)*

---------

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Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

Author: github-actions[bot]

plots/phase-diagram/implementations/pygal.py

@@ -0,0 +1,106 @@
+""" pyplots.ai
+phase-diagram: Phase Diagram (State Space Plot)
+Library: pygal 3.1.0 | Python 3.13.11
+Quality: 91/100 | Created: 2025-12-31
+"""
+
+import numpy as np
+import pygal
+from pygal.style import Style
+
+
+# Data: Damped pendulum phase space
+# The damped pendulum follows: d²x/dt² = -ω²x - γ(dx/dt)
+# This creates a spiral trajectory converging to equilibrium
+
+np.random.seed(42)
+
+# Parameters for damped harmonic oscillator
+omega = 2.0  # Natural frequency
+gamma = 0.3  # Damping coefficient
+dt = 0.02  # Time step
+t_max = 15  # Total time
+
+# Generate trajectory using numerical integration (Euler method)
+t = np.arange(0, t_max, dt)
+n_points = len(t)
+
+# Initial conditions: displaced position, zero velocity
+x = np.zeros(n_points)
+v = np.zeros(n_points)  # v = dx/dt
+x[0] = 2.0  # Initial displacement
+v[0] = 0.0  # Initial velocity
+
+# Integrate the equations of motion
+for i in range(1, n_points):
+    # Acceleration: a = -ω²x - γv
+    a = -(omega**2) * x[i - 1] - gamma * v[i - 1]
+    # Update velocity and position
+    v[i] = v[i - 1] + a * dt
+    x[i] = x[i - 1] + v[i] * dt
+
+# Also generate an undamped oscillator for comparison (limit cycle)
+x_undamped = np.zeros(n_points)
+v_undamped = np.zeros(n_points)
+x_undamped[0] = 1.5
+v_undamped[0] = 0.0
+
+for i in range(1, n_points):
+    a = -(omega**2) * x_undamped[i - 1]
+    v_undamped[i] = v_undamped[i - 1] + a * dt
+    x_undamped[i] = x_undamped[i - 1] + v_undamped[i] * dt
+
+# Custom style for large canvas
+custom_style = Style(
+    background="white",
+    plot_background="white",
+    foreground="#333333",
+    foreground_strong="#333333",
+    foreground_subtle="#666666",
+    colors=("#306998", "#FFD43B", "#FF6B6B"),
+    title_font_size=60,
+    label_font_size=40,
+    major_label_font_size=36,
+    legend_font_size=36,
+    value_font_size=28,
+    stroke_width=4,
+    opacity=0.85,
+    opacity_hover=1.0,
+)
+
+# Create XY chart (scatter plot with lines)
+chart = pygal.XY(
+    width=4800,
+    height=2700,
+    style=custom_style,
+    title="phase-diagram · pygal · pyplots.ai",
+    x_title="Position x",
+    y_title="Velocity dx/dt",
+    show_dots=False,  # Lines only for smooth trajectory
+    stroke=True,
+    fill=False,
+    show_x_guides=True,
+    show_y_guides=True,
+    dots_size=3,
+    stroke_style={"width": 4},
+    legend_at_bottom=True,
+    legend_box_size=24,
+    truncate_legend=-1,
+    show_legend=True,
+)
+
+# Downsample for visualization (pygal works better with fewer points)
+step = 3
+damped_points = [(float(x[i]), float(v[i])) for i in range(0, n_points, step)]
+undamped_points = [(float(x_undamped[i]), float(v_undamped[i])) for i in range(0, n_points, step)]
+
+# Add trajectories
+chart.add("Damped Oscillator (γ=0.3)", damped_points)
+chart.add("Undamped Oscillator (Limit Cycle)", undamped_points)
+
+# Add fixed point marker at origin
+chart.add("Equilibrium (Fixed Point)", [(0, 0)], dots_size=12, show_dots=True, stroke=False)
+
+# Save outputs
+chart.render_to_file("plot.html")
+chart.render_to_png("plot.png")

plots/phase-diagram/metadata/pygal.yaml

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+library: pygal
+specification_id: phase-diagram
+created: '2025-12-31T11:12:39Z'
+updated: '2025-12-31T11:20:54Z'
+generated_by: claude-opus-4-5-20251101
+workflow_run: 20617709332
+issue: 3004
+python_version: 3.13.11
+library_version: 3.1.0
+preview_url: https://storage.googleapis.com/pyplots-images/plots/phase-diagram/pygal/plot.png
+preview_thumb: https://storage.googleapis.com/pyplots-images/plots/phase-diagram/pygal/plot_thumb.png
+preview_html: https://storage.googleapis.com/pyplots-images/plots/phase-diagram/pygal/plot.html
+quality_score: 91
+review:
+  strengths:
+  - Excellent demonstration of phase space concepts with both damped and undamped
+    trajectories
+  - Clean visual distinction between the spiral (damped) and closed loop (limit cycle)
+    behaviors
+  - Well-chosen colorblind-safe color scheme (blue and yellow)
+  - Proper mathematical implementation of damped harmonic oscillator using Euler integration
+  - Good use of pygal custom styling for large canvas rendering
+  weaknesses:
+  - The equilibrium point marker at origin is quite small and could be more prominent
+  - The undamped oscillator trajectory shows some numerical drift (not a perfect ellipse
+    due to Euler method)