How to make a cat map? - briefly

To create a cat map, you need to understand the concept of chaotic dynamics and apply it using linear algebra. Specifically, you will use a matrix with integer entries that represent the transformation on the torus, where the cat's position is mapped onto itself in a nonlinear manner.

How to make a cat map? - in detail

Creating a cat map is an intriguing process that involves understanding the mathematical principles behind it and applying them systematically. A cat map, also known as the Arnold's cat map, is a simple example of chaotic dynamics in mathematics. Here’s a step-by-step guide to creating one:

1. Understand the Mathematical Basis: The cat map is defined by a linear transformation on the torus (a doughnut shape). Specifically, it is represented by the matrix: [ \begin{pmatrix} 2 & 1 \ 3 & 2 \end{pmatrix} ] This matrix acts on the coordinates of a point on the torus.

2. Initial Setup: Begin with an initial point ((x, y)) in the unit square ([0, 1]^2). In this context, points at the boundary are considered equivalent to those on the opposite side due to the toroidal nature of the map. For example, ((1, y)) is the same as ((0, y)).

3. Apply the Transformation: To find the image of a point under the cat map, multiply the matrix by the coordinate vector: [ \begin{pmatrix} 2 & 1 \ 3 & 2 \end{pmatrix} \cdot \begin{pmatrix} x \ y \end{pmatrix}

   \begin{pmatrix} 2x + y \ 3x + 2y \end{pmatrix} ]

4. Modulo Operation: Since the map is defined on a torus, we need to take the results modulo 1: [ x' = (2x + y) \mod 1 ] [ y' = (3x + 2y) \mod 1 ] This ensures that the new coordinates fall within the unit square.

5. Iterate the Process: Repeat the transformation for multiple iterations to visualize how the point evolves over time. Each iteration represents a step forward in time, demonstrating the chaotic nature of the map.

6. Visualization: Plotting the trajectory of an initial point under repeated applications of the cat map provides a visual representation of the dynamics. This is often done using a computer program to handle the iterations and plotting efficiently.

7. Software Implementation: For practical purposes, writing a simple script in Python or another programming language can help automate the process:

   import numpy as np
   import matplotlib.pyplot as plt
   def cat_map(x, y, n):
    xs, ys = [x], [y]
    for _ in range(n):
    x, y = (2*x + y) % 1, (3*x + 2*y) % 1
    xs.append(x)
    ys.append(y)
    return xs, ys
   initial_point = (0.5, 0.3)
   iterations = 100
   xs, ys = cat_map(*initial_point, iterations)
   plt.plot(xs, ys)
   plt.xlabel('x')
   plt.ylabel('y')
   plt.title('Cat Map Trajectory')
   plt.show()

By following these steps, you can create and visualize a cat map, exploring the fascinating world of chaotic dynamics.