产生逻辑图的分叉图，包括不稳定点

Question

典型分叉图仅包含稳定的固定点，这些点很有吸引力（附近的点在迭代时被拉到固定点）。这些非常容易计算：只需迭代一遍次，它会收敛到固定点。 但是，我试图包括不稳定的（排斥）固定点。从本质上讲，不会使用上述算法找到它们。 我的想法是松开要固定点的条件。也就是说，如果您的x值在f（x）的“ epsilon”之内，则只需返回x即可。这不会产生所需的输出。 我在下面包含了我的代码。任何想法或反馈都将不胜感激！ 谢谢

import numpy as np
from matplotlib import pyplot as plt

# Define the logistic map
epsilon = 1e-4
def lmap(x, r):
  # "Beefing up" the fixed point: 
  # if x is in the epsilon neighbourhood of f(x), let it be a fixed point (return x)
  if abs(r * x * (1 - x) - x) < epsilon:
      return x
  else:
      return r * x * (1 - x)
    
# Iterate over the logistic map to find fixed points
def get_fixed_pts(f, x0, r, iterations = 100, keep = 10, decimals = 5):
  x = x0

  for i in range(iterations):
    x = f(x, r)
    if i >= (iterations - keep):
        yield np.round(x, decimals)

# Setting up some params
x0 = 0.5
n_pts = 1000
r_lims = [2, 4]

fixed_points = []
corresponding_r_values = []

# Looping over r values to find the fixed point for every r value
for r in np.linspace(*r_lims, n_pts):
  # Get the fixed points and remove duplicates. 
  # Then, find the cycle size (number of fixed points)
  pts = set(get_fixed_pts(lmap, x0, r))
  cycle_size = len(pts)

  # Store values
  fixed_points.extend(pts)
  corresponding_r_values.extend([r] * cycle_size)

# Plot all found points
plt.figure(figsize=(16, 9))
plt.scatter(corresponding_r_values, fixed_points, s = 0.01, c = 'black', label = 'Stable fixed point')
plt.title('Bifurcation diagram of logistic map')
plt.xlabel('Parameter Value')
plt.ylabel('Fixed Points')
plt.xlim(r_lims)



plt.legend()
plt.show()

要用不稳定的固定点生成逻辑图的分叉图，您可以通过分析计算固定点，因为不稳定点驱动轨迹。

使用f（x）= r * x *（1 -x）查找固定点x = 0和x =（r -1）/r，

并将它们分类为稳定| f'（x）|

1使用衍生f'（x）= r *（1-2x）

import numpy as np
import matplotlib.pyplot as plt

def logistic_map(x, r):
    return r * x * (1 - x)

def logistic_map_derivative(x, r):
    return r * (1 - 2 * x)

def compute_fixed_points(r):
    if r < 1:
        return [0], []
    stable = []
    unstable = []
    if abs(logistic_map_derivative(0, r)) < 1:
        stable.append(0)
    else:
        unstable.append(0)
    if r > 1:
        x_fixed = (r - 1) / r
        if abs(logistic_map_derivative(x_fixed, r)) < 1:
            stable.append(x_fixed)
        else:
            unstable.append(x_fixed)
    return stable, unstable

# Iterate through the logistic map to compute periodic points
def iterate_logistic_map(x0, r, iterations=500, discard=100):
    x = x0
    trajectory = []
    for i in range(iterations):
        x = logistic_map(x, r)
        if i >= discard:
            trajectory.append(x)
    return trajectory

r_values = np.linspace(2, 4, 1000)
x0 = 0.5
num_iterations = 500
discard = 100

stable_points = []
unstable_points = []
r_stable = []
r_unstable = []

for r in r_values:
    # Compute analytically derived fixed points
    stable, unstable = compute_fixed_points(r)
    stable_points.extend(stable)
    unstable_points.extend(unstable)
    r_stable.extend([r] * len(stable))
    r_unstable.extend([r] * len(unstable))
    # Iterate to find attracting fixed points
    trajectory = iterate_logistic_map(x0, r, iterations=num_iterations, discard=discard)
    r_stable.extend([r] * len(trajectory))
    stable_points.extend(trajectory)

plt.figure(figsize=(16, 9))
plt.scatter(r_stable, stable_points, s=0.5, c='black', label='Stable Points')
plt.scatter(r_unstable, unstable_points, s=1, c='red', label='Unstable Points')
plt.title('Bifurcation Diagram of Logistic Map (Stable and Unstable Points)')
plt.xlabel('Parameter r')
plt.ylabel('x')
plt.legend(loc='upper left')
plt.show()