The construction method of chaotic system model based on state variables and uncertain variables and its application in image encryption

Abstract

This paper focuses on the construction of nonlinear dynamic models, specifically targeting continuous chaotic systems. It introduces an innovative approach to integrating state variables and uncertain variables to construct continuous chaotic systems. Initially, a unified construction method is proposed, combining state variables with a determinable amplitude matrix. The feasibility of this method is verified through the definition of the Lyapunov index. Using a three-dimensional chaotic system with a single nonlinear structure as an example, numerical parameters are screened based on the Lyapunov index. Additionally, permutation entropy is utilized to explore changes in the system's chaotic behavior induced by random natural values and mathematical functions in random amplitudes. The model is then extended to a four-dimensional state, demonstrating the complex characteristics of the system in this higher dimension through various complexities, thus highlighting the universal applicability of the construction method. Finally, the constructed three-dimensional and four-dimensional continuous systems are applied to a DNA-based color image encryption system. Comparative analyses of robustness and security show that the proposed method offers significant advantages in practical applications. The results indicate that this method is not only theoretically innovative in constructing chaotic systems but also holds high practical value.