feat(pygal): implement bifurcation-basic (#5107)

## Implementation: `bifurcation-basic` - pygal

Implements the **pygal** version of `bifurcation-basic`.

**File:** `plots/bifurcation-basic/implementations/pygal.py`

**Parent Issue:** #4415

---
:robot: *[impl-generate
workflow](https://github.com/MarkusNeusinger/pyplots/actions/runs/23361206415)*

---------

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

Author: github-actions[bot]

plots/bifurcation-basic/implementations/pygal.py

@@ -0,0 +1,171 @@
+""" pyplots.ai
+bifurcation-basic: Bifurcation Diagram for Dynamical Systems
+Library: pygal 3.1.0 | Python 3.14.3
+Quality: 90/100 | Created: 2026-03-20
+"""
+
+import numpy as np
+import pygal
+from pygal.style import Style
+
+
+# Data — logistic map x(n+1) = r * x(n) * (1 - x(n))
+np.random.seed(42)
+transient = 200
+iterations = 100
+x0 = 0.1 + np.random.uniform(-0.01, 0.01)
+
+# Key bifurcation thresholds
+R_PERIOD2 = 3.0
+R_PERIOD4 = 3.449
+R_PERIOD8 = 3.544
+R_CHAOS = 3.57
+
+# Variable-density sampling: more points in complex regions
+r_stable = np.linspace(2.5, R_PERIOD2, 250)
+r_periodic = np.linspace(R_PERIOD2, R_CHAOS, 500)
+r_chaotic = np.linspace(R_CHAOS, 4.0, 700)
+r_values = np.concatenate([r_stable, r_periodic, r_chaotic])
+
+# Colorblind-safe palette: navy blue, burnt orange, deep violet (no blue-green confusion)
+regions = {
+    "Stable Fixed Point": (2.5, R_PERIOD2, "#1b5e8a"),
+    "Period-Doubling Cascade": (R_PERIOD2, R_CHAOS, "#d55e00"),
+    "Chaotic Regime": (R_CHAOS, 4.0, "#7b2d8e"),
+}
+
+region_data = {name: [] for name in regions}
+
+for r in r_values:
+    x = x0
+    for _ in range(transient):
+        x = r * x * (1.0 - x)
+    for _ in range(iterations):
+        x = r * x * (1.0 - x)
+        for name, (lo, hi, _) in regions.items():
+            if lo <= r < hi or (name == "Chaotic Regime" and r == 4.0):
+                region_data[name].append(
+                    {"value": (round(float(r), 5), round(float(x), 5)), "label": f"r={r:.4f}, x={x:.4f}"}
+                )
+                break
+
+# Downsample each region to balance visual density
+max_per_region = {"Stable Fixed Point": 6000, "Period-Doubling Cascade": 18000, "Chaotic Regime": 28000}
+for name in region_data:
+    pts = region_data[name]
+    cap = max_per_region[name]
+    if len(pts) > cap:
+        idx = np.random.choice(len(pts), cap, replace=False)
+        idx.sort()
+        region_data[name] = [pts[i] for i in idx]
+
+# Publication-quality style with high-contrast colorblind-safe palette
+font = "'Helvetica Neue', 'DejaVu Sans', Helvetica, Arial, sans-serif"
+region_colors = tuple(c for _, (_, _, c) in regions.items())
+annotation_color = "#888888"
+all_colors = region_colors + (annotation_color,)
+
+custom_style = Style(
+    background="white",
+    plot_background="#f7f7f7",
+    foreground="#333333",
+    foreground_strong="#111111",
+    foreground_subtle="#dddddd",
+    guide_stroke_color="#e0e0e0",
+    guide_stroke_dasharray="3, 8",
+    major_guide_stroke_dasharray="2, 4",
+    colors=all_colors,
+    font_family=font,
+    title_font_family=font,
+    title_font_size=52,
+    label_font_size=40,
+    major_label_font_size=36,
+    legend_font_size=30,
+    legend_font_family=font,
+    value_font_size=26,
+    tooltip_font_size=28,
+    tooltip_font_family=font,
+    opacity=0.55,
+    opacity_hover=1.0,
+)
+
+# Chart with pygal-specific features: secondary series, custom formatters, interpolation config
+chart = pygal.XY(
+    width=4800,
+    height=2700,
+    style=custom_style,
+    title="bifurcation-basic · pygal · pyplots.ai",
+    x_title="Growth Rate Parameter (r)",
+    y_title="Steady-State Population (xₙ)",
+    show_legend=True,
+    legend_at_bottom=True,
+    legend_at_bottom_columns=4,
+    legend_box_size=22,
+    stroke=False,
+    dots_size=1.8,
+    show_x_guides=True,
+    show_y_guides=True,
+    show_y_minor_guides=True,
+    x_value_formatter=lambda v: f"{v:.3f}",
+    value_formatter=lambda v: f"{v:.4f}",
+    margin_bottom=110,
+    margin_left=70,
+    margin_right=50,
+    margin_top=55,
+    xrange=(2.5, 4.0),
+    range=(0.0, 1.0),
+    print_values=False,
+    print_zeroes=False,
+    js=[],
+    x_labels=[2.5, R_PERIOD2, 3.2, R_PERIOD4, R_PERIOD8, 3.7, 3.8, 4.0],
+    x_labels_major=[R_PERIOD2, R_PERIOD4, R_PERIOD8],
+    y_labels=[0.0, 0.2, 0.4, 0.6, 0.8, 1.0],
+    truncate_legend=-1,
+    no_data_text="",
+    show_x_labels=True,
+    show_y_labels=True,
+    dynamic_print_values=True,
+    allow_interruptions=True,
+    show_minor_x_labels=True,
+    spacing=25,
+    inner_radius=0,
+    include_x_axis=True,
+)
+
+# Add each region as a separate series with per-point tooltip metadata
+for name in regions:
+    lo, hi, _ = regions[name]
+    chart.add(
+        f"{name} (r\u2248{lo:.1f}\u2013{hi:.2f})",
+        region_data[name],
+        stroke=False,
+        show_dots=True,
+        allow_interruptions=True,
+    )
+
+# Annotation markers at key bifurcation points — dashed vertical lines in one legend entry
+annotation_points = [
+    (R_PERIOD2, "r\u22483.0: Period-2 onset"),
+    (R_PERIOD4, "r\u22483.449: Period-4 onset"),
+    (R_PERIOD8, "r\u22483.544: Period-8 onset"),
+]
+
+annotation_data = []
+for r_val, label in annotation_points:
+    annotation_data.append({"value": (r_val, 0.0), "label": label})
+    annotation_data.append({"value": (r_val, 1.0), "label": label})
+    annotation_data.append(None)
+
+chart.add(
+    "Bifurcation Points",
+    annotation_data,
+    stroke=True,
+    stroke_style={"width": 2.5, "dasharray": "10, 5"},
+    show_dots=False,
+    dots_size=0,
+    secondary=True,
+)
+
+# Dual render: PNG for static preview, HTML for pygal's native SVG interactivity with tooltips
+chart.render_to_png("plot.png")
+chart.render_to_file("plot.html")