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Hénon Map

Overview

The Hénon map is a discrete-time dynamical system that serves as a simple model of chaotic systems. It is defined by a two-dimensional recursive relation:

\[ \begin{align*} x_{n+1} &= 1 - a x_n^2 + y_n \\ y_{n+1} &= b x_n \end{align*} \]

Where: - \(a\) and \(b\) are parameters that control the behavior of the system. - \((x_n, y_n)\) represents the state of the system at iteration \(n\).

For the classic values \(a = 1.4\) and \(b = 0.3\), the map exhibits chaotic behavior and forms a strange attractor.

Python Code Example: Simulating and Visualizing the Hénon Map

import numpy as np
import matplotlib.pyplot as plt

# Define the Hénon map
def henon_map(x, y, a, b):
    x_next = 1 - a * x**2 + y
    y_next = b * x
    return x_next, y_next

# Parameters
a = 1.4
b = 0.3

# Initial conditions
x, y = 0.0, 0.0

# Number of iterations
iterations = 10000

# Store the results
x_values = []
y_values = []

# Iterate the map
for _ in range(iterations):
    x, y = henon_map(x, y, a, b)
    x_values.append(x)
    y_values.append(y)

# Plot the Hénon attractor
plt.figure(figsize=(8, 8))
plt.scatter(x_values, y_values, s=0.1, color="purple")
plt.title("Hénon Map Attractor")
plt.xlabel("x")
plt.ylabel("y")
plt.grid()
plt.savefig("docs/Mechanics/pic/henon_map.png")
plt.show()

Key Insights

  1. Chaotic Behavior: The Hénon map is one of the simplest systems to exhibit chaos, making it a foundational model in dynamical systems.

  2. Strange Attractor: The attractor forms a fractal structure, highlighting the self-similar nature of chaotic systems.

  3. Parameter Sensitivity: Small changes in \(a\) or \(b\) can lead to significant differences in the attractor's shape and dynamics.

Suggested Projects

  1. Parameter Exploration: Investigate how varying \(a\) and \(b\) affects the attractor's shape and stability.

  2. Fractal Dimension: Calculate the fractal dimension of the Hénon attractor to quantify its complexity.

  3. Lyapunov Exponent: Compute the Lyapunov exponent to characterize the map's chaotic nature.

  4. 3D Visualization: Extend the analysis to include a time dimension or color code the points by iteration.

This example introduces the chaotic dynamics of the Hénon map. Experiment with different initial conditions and parameters to uncover the fascinating behavior of this simple yet rich system!