Multi-Valued Model for Generating Complex Chaos and Fractals

Designing chaotic maps and fractal maps with rich dynamics in the complex field presents an interesting and challenging research topic. In this paper, we propose a novel approach called the one-dimensional multi-valued model (1D-MVM) for generating 1D complex chaotic and fractal maps by levering both single-valued and multi-valued functions. To demonstrate the effectiveness of the 1D-MVM, we present two 1D complex chaotic maps and one 1D fractal map as specific examples. Theoretical analysis confirms that the chaotic maps generated by the 1D-MVM exhibit chaotic behavior, while property analysis reveals that these chaotic maps possess hyperchaotic strange attractors, with their associated Lyapunov exponents being determined by specific system parameters. We also conduct extensive experiments to demonstrate the intricate dynamics and high performance indicators of the newly generated complex chaotic maps. A hardware platform is developed and the principal value attractors of these chaotic maps are experimentally captured. In addition, we explore the application of the hyperchaotic sequences generated by these complex chaotic maps to the pseudo-random number generators. Rigorous testing results validate the high degree of randomness exhibited by the generated pseudo-random numbers. Finally, we leverage the second branch of the 1D fractal map to generate a diverse range of fractal structures, further demonstrating the versatility and potential applications of the proposed approach.