Extend outer mu branches so the trajectory reaches the centre

x(mu) only approaches the centre as |mu| -> inf, so the sampling in
[mean(poles) - 3, mean(poles) + 3] left a gap around it. The outer
branches now continue geometrically up to |mu| ~ 1e4 / lambda_min; the
extra samples lie outside the mu axes and only show in the trajectory.

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>

plot_mu_trajectories.py

@@ -26,6 +26,7 @@ CASES = {
 }
 
 N_PER_BRANCH = 266    # mu samples between consecutive poles
+N_TAIL = 100          # extra samples extending the outer branches geometrically up to |mu| ~ 1e4 / lambda_min
 EPS = 0.01            # gap left on each side of a pole
 MU_HALF_WIDTH = 3     # mu is sampled in [m - MU_HALF_WIDTH, m + MU_HALF_WIDTH], m the mean of the poles
 N_DOTTED_OUTER = 2    # points drawn dotted at the ends of the outer branches
@@ -68,11 +69,19 @@ def plot_trajectories(Q, x0):
 
     mu_min = poles.mean() - MU_HALF_WIDTH
     mu_max = poles.mean() + MU_HALF_WIDTH
-    mu = np.concatenate([np.linspace(lo, hi, N_PER_BRANCH) for lo, hi in
+    # x(mu) only reaches the centre as |mu| -> inf, so the outer branches continue geometrically
-                         [(mu_min, poles[0] - EPS), (poles[0] + EPS, poles[1] - EPS), (poles[1] + EPS, mu_max)]])
+    # beyond [mu_min, mu_max]; these samples show up in the trajectory but not on the mu axes.
+    tail = np.geomspace(MU_HALF_WIDTH / N_PER_BRANCH, 1e4 / Q.eig.min(), N_TAIL)
+    branch_samples = [
+        np.concatenate((mu_min - tail[::-1], np.linspace(mu_min, poles[0] - EPS, N_PER_BRANCH))),
+        np.linspace(poles[0] + EPS, poles[1] - EPS, N_PER_BRANCH),
+        np.concatenate((np.linspace(poles[1] + EPS, mu_max, N_PER_BRANCH), mu_max + tail)),
+    ]
+    mu = np.concatenate(branch_samples)
     idx = np.arange(len(mu))
+    pole_indices = np.cumsum([len(samples) for samples in branch_samples[:-1]])
     # A pole where x0 has a zero standardized component vanishes and joins its neighbouring branches.
-    branches = np.split(idx, [k * N_PER_BRANCH for k in (1, 2) if pole_active[k - 1]])
+    branches = np.split(idx, pole_indices[pole_active])
 
     cmap = colormaps[COLORMAP]
     norm = Normalize(0, len(mu) - 1)
@@ -115,8 +124,8 @@ def plot_trajectories(Q, x0):
         ax_x.scatter(*Q.x(root, x0), **style)
 
     if not degenerate:
-        for k, label in zip((1, 2), POLE_LABELS):
+        for pole_index, label in zip(pole_indices, POLE_LABELS):
-            for i in (k * N_PER_BRANCH - 1, k * N_PER_BRANCH):
+            for i in (pole_index - 1, pole_index):
                 ax_x.text(*xs[i], label, **TEXT_STYLE)
         for k in range(Q.dim):
             point, direction = Q.pole_line(x0, k)