How to draw a cat map? - briefly
To create a cat map, start by sketching a simple, rounded triangle for the cat's head and body. Add facial features such as eyes, a nose, and a mouth, then draw ears at the top and a tail extending from the back.
How to draw a cat map? - in detail
Drawing a cat map, also known as the cat transformation, is a fascinating exercise in chaos theory and dynamical systems. This map is a discrete chaotic map that exhibits complex behavior despite its simplicity. To draw a cat map, follow these detailed steps:
First, understand the mathematical foundation. The cat map is defined by a linear transformation of a point (x, y) on a 2D torus. The transformation is given by the matrix equation:
[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 1 & 2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \mod 1 ]
Here, (x', y') are the new coordinates after the transformation, and the operation mod 1 ensures that the points wrap around the torus.
To visualize this transformation, you need to iteratively apply the matrix to an initial set of points. Start by selecting a grid of points on a unit square, where each point represents a coordinate (x, y). For example, you can use a 100x100 grid for a detailed visualization.
Next, apply the transformation matrix to each point in the grid. Compute the new coordinates (x', y') using the matrix equation provided. Since the coordinates must remain within the unit square, take the fractional part of the results using the mod 1 operation. This step ensures that the points wrap around the edges of the square, simulating the behavior on a torus.
To draw the cat map, plot the initial points and the transformed points in the same coordinate system. Use different colors or styles to distinguish between the initial and transformed points. This will help you observe the chaotic behavior of the map.
For a more dynamic visualization, iterate the transformation multiple times. After each iteration, apply the transformation matrix to the new set of points and plot the results. Observe how the points spread out and fill the unit square in a seemingly random manner. This demonstrates the mixing property of the cat map, where nearby points quickly diverge and fill the space uniformly.
Additionally, you can experiment with different initial conditions and grid sizes to explore the sensitivity of the cat map to initial conditions. This is a hallmark of chaotic systems, where small changes in initial conditions lead to drastically different outcomes over time.
To enhance the visualization, consider using computational tools such as Python with libraries like Matplotlib or NumPy. These tools allow for efficient computation and plotting of the cat map, making it easier to observe the complex behavior.
In summary, drawing a cat map involves understanding the linear transformation defined by the matrix, iterating the transformation on a grid of points, and visualizing the results. This process reveals the chaotic and mixing properties of the cat map, providing insights into the behavior of dynamical systems.