update

Source/TODEL_area.py

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+import numpy as np
+import matplotlib.pyplot as plt
+import potentials as pot
+import integrator as itg
+import initial_conditions as init
+
+# Parameters
+h = 0.01  # Step size for integrator
+N_inter = 500 # Number of iterations for distance calculation
+eps = 1e-3  # Perturbation distance
+mu_c = 0.6  # Threshold for classifying ergodic vs regular
+energy_values = np.array([0.01, 0.02, 0.04, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18])
+N_PART = 100
+
+def classify_point(P1, P1_prime):
+    pos_t1, positions1 = itg.rk4(0, P1, h, N_inter, pot.hh_evolution)
+    pos_t2, positions2 = itg.rk4(0, P1_prime, h, N_inter, pot.hh_evolution)
+
+    # Compute the sum of squared distances (μ)
+    mu = 0
+    for i in range(N_inter):
+        y1, y_dot1 = positions1[i, 0, 1], positions1[i, 1, 1]
+        y2, y_dot2 = positions2[i, 0, 1], positions2[i, 1, 1]
+        x1, x_dot1 = positions1[i, 0, 0], positions1[i, 1, 0]
+        x2, x_dot2 = positions2[i, 0, 0], positions2[i, 1, 0]
+
+        squared_distance = (np.sqrt((x2 - x1)**2 + (y2 - y1) ** 2 + (x_dot2 - x_dot1) ** 2 +(y_dot2 - y_dot1) ** 2))**2
+        mu += squared_distance
+
+    return mu
+
+# Analyze for different energy values
+relative_areas = []
+
+for E in energy_values:
+    # Generate initial conditions for the given energy level
+    initial_conditions = init.n_energy_part(pot.hh_potential, N=N_PART, E=E)
+    initial_conditions_eps = initial_conditions.copy()
+    initial_conditions_eps[0,1] += eps
+    n_regular = 0
+
+    # Loop through each initial condition
+    for i in range(N_PART):
+        P1 = initial_conditions[:, :, i]
+        # Perturb the initial point slightly
+        P1_prime = initial_conditions_eps[:,:,i]
+
+        # Classify the point based on the evolution
+        mu = classify_point(P1, P1_prime)
+        np.nan_to_num(mu, copy=False, nan=1.0)
+
+        print(f"the mu value is {mu}")
+        if mu < mu_c:
+            n_regular += 1
+
+    # Compute the relative area covered by regular points
+    relative_area = n_regular / N_PART
+    relative_areas.append(relative_area)
+
+# Plot the results
+plt.figure()
+plt.plot(energy_values, relative_areas, marker='o', color='C0')
+plt.xlabel("Energy (E)")
+plt.ylabel("Relative Area Covered by Regular Curves")
+plt.title("Transition from Ergodic to Stable Regime")
+plt.grid()
+plt.show()