Design of Multi-Cavity Chaotic Maps in the Polar Coordinate Using Nonlinear Curves and Modulo Operation with Application to Cavity-Based Data Hiding

At present, constructing discrete chaotic systems with unique characteristics and chaos has become a focal topic in the field of nonlinear research. This paper presents a new framework for designing multi-cavity chaotic maps in polar coordinates. It constructs the basic chaotic map through nonlinear curves (such as Lotus curve, rose curves, and star curves) and generates multi-cavity attractors based on modular arithmetic. The nonlinear curves introduce complex deformations in the angular and radial components, while modular arithmetic serves as a folding mechanism to confine the dynamics to a specific range. The combined effect of these two elements forms multiple clearly separated chaotic cavities in the phase space. The number, size, shape, and chaotic characteristics of the cavities can be flexibly controlled by parameters. However, the introduction of fractional-order difference operators will disrupt the multi-chamber structure and make the system more complex. Furthermore, a data-hiding scheme based on the cavities is developed: the cavities act as natural isolation containers to embed information bits, and the chaotic dynamics provide encryption and confusion mechanisms. Experiments show that the designed chaotic map has high complexity and rich bifurcation behaviors; the data-hiding scheme performs well in terms of embedding capacity and security.