On the construction of one-dimensional discrete chaos theory based on the improved version of Marotto’s theorem

• Published in: Journal of computational and applied mathematics 2020-12, Vol.380, p.112952, Article 112952
• Main Authors: Li, Jiu, Zang, Hongyan, Wei, Xinyuan
• Format: Article
• Language: English
• Subjects: Chaotic system; Lyapunov exponent; Marotto’s theorem; Polynomial chaos; Snap-back repeller
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2020.112952

The purpose of this paper is to design one-dimensional chaotic maps with a non-zero intercept in the Y-axis and research the basic dynamic characteristics. First, by studying the common conditions that make one-dimensional discrete dynamical systems chaotic under the modulo operation, the general theory of the chaotic behavior of one-dimensional nonlinear functions with a non-zero origin is presented. It is proved that the discrete dynamic system derived from the proposed theory exhibits chaos phenomena in the sense of Li-Yorke, utilizing the improved version of Marotto’s theorem. Furthermore, the part of the discrete dynamical functions satisfying the proposed chaos theory is given. The function track, bifurcation diagram and Lyapunov exponential spectrum are also analyzed. Numerical simulations agree with the analytical framework for the suggested chaos theory. Finally, given the application of chaos in cryptographic communication, two significant corollaries of the original functions as general polynomials are put forward according to the theory of chaos structure.