Bifurcation diagram with python

When it comes to creating a bifurcation diagram with Python, there are several approaches you can take. In this article, we will explore three different ways to solve this problem, each with its own advantages and disadvantages.

Option 1: Using Matplotlib

One way to create a bifurcation diagram is by using the Matplotlib library in Python. Matplotlib is a powerful plotting library that allows you to create various types of visualizations, including line plots, scatter plots, and histograms.

import matplotlib.pyplot as plt

def bifurcation_diagram(x0, r_values, n_iterations):
    x = x0
    results = []
    
    for r in r_values:
        for _ in range(n_iterations):
            x = r * x * (1 - x)
        results.append(x)
    
    plt.plot(r_values, results, 'k.', markersize=1)
    plt.xlabel('r')
    plt.ylabel('x')
    plt.title('Bifurcation Diagram')
    plt.show()

x0 = 0.5
r_values = [2.4, 3.5, 3.8, 3.9, 4.0]
n_iterations = 1000

bifurcation_diagram(x0, r_values, n_iterations)

This code defines a function bifurcation_diagram that takes in the initial value x0, a list of r_values to iterate over, and the number of n_iterations to perform. It then calculates the bifurcation points and plots them using Matplotlib.

Option 2: Using NumPy

Another approach is to use the NumPy library, which provides support for large, multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays.

import numpy as np
import matplotlib.pyplot as plt

def bifurcation_diagram(x0, r_values, n_iterations):
    x = x0
    results = []
    
    for r in r_values:
        for _ in range(n_iterations):
            x = r * x * (1 - x)
        results.append(x)
    
    plt.plot(r_values, results, 'k.', markersize=1)
    plt.xlabel('r')
    plt.ylabel('x')
    plt.title('Bifurcation Diagram')
    plt.show()

x0 = 0.5
r_values = np.linspace(2.4, 4.0, 1000)
n_iterations = 1000

bifurcation_diagram(x0, r_values, n_iterations)

In this code, we import the NumPy library and use the np.linspace function to generate a linearly spaced array of r_values between 2.4 and 4.0. This allows us to have a more precise representation of the bifurcation diagram.

Option 3: Using Pandas

The third option involves using the Pandas library, which provides data manipulation and analysis tools. While Pandas is primarily used for working with tabular data, it can also be used to create visualizations.

import pandas as pd
import matplotlib.pyplot as plt

def bifurcation_diagram(x0, r_values, n_iterations):
    x = x0
    results = []
    
    for r in r_values:
        for _ in range(n_iterations):
            x = r * x * (1 - x)
        results.append(x)
    
    data = pd.DataFrame({'r': r_values, 'x': results})
    data.plot(x='r', y='x', kind='scatter', s=1)
    plt.xlabel('r')
    plt.ylabel('x')
    plt.title('Bifurcation Diagram')
    plt.show()

x0 = 0.5
r_values = np.linspace(2.4, 4.0, 1000)
n_iterations = 1000

bifurcation_diagram(x0, r_values, n_iterations)

In this code, we import the Pandas library and create a DataFrame using the pd.DataFrame function. We then use the data.plot method to create a scatter plot of the bifurcation points.

After exploring these three options, it is clear that using Matplotlib is the most straightforward and efficient way to create a bifurcation diagram in Python. It provides a wide range of customization options and is widely used in the scientific community for data visualization. However, if you are already familiar with NumPy or Pandas, you can also achieve the desired result using these libraries.

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