MATLAB Implementation of 3D Baker Map with Chaotic Dynamics Analysis

The Baker Map is a classical chaotic system model extensively used for studying nonlinear dynamics and chaotic behavior. Its fundamental principle involves simulating dynamic processes under chaotic conditions through stretching and folding operations. The 3D version of Baker Map extends the traditional two-dimensional model by incorporating additional dimensions, resulting in more complex dynamic characteristics. Implementing the 3D Baker Map in MATLAB typically requires the following steps: First, define initial conditions, usually as point sets or grids in three-dimensional space. Then perform iterative calculations according to the mathematical rules of Baker Map, incorporating stretching and folding operations. Finally, visualize the results to observe chaotic behavior patterns. Key implementation aspects include using meshgrid() for initial 3D point generation and implementing efficient iteration through matrix operations. The core of the 3D Baker Map lies in its iterative formula, which determines the new position of each point after every iteration. Unlike the 2D version, the 3D Baker Map requires additional transformations along the z-axis, typically involving layering, compression, and displacement operations. The mathematical formulation can be implemented using conditional statements and coordinate transformations, where each dimension undergoes specific scaling and folding rules. Through multiple iterations, initially distributed points gradually diffuse and form complex chaotic attractor structures. To enhance readability and computational efficiency, MATLAB implementations can utilize vectorized operations instead of loops, significantly improving performance for large point sets. Built-in 3D visualization functions like scatter3() or plot3() can be employed for effective result display. Code optimization may include preallocating arrays for trajectory storage and using quiver3() for vector field representation. Furthermore, different dynamic behavior patterns can be explored by adjusting parameters such as stretching factors or folding methods, implemented through customizable input parameters and modular function design.