n-Dimensional Polynomial Chaotic System With Applications

• Published in: IEEE transactions on circuits and systems. I, Regular papers Vol. 69; no. 2; pp. 784 - 797
• Main Authors: Hua, Zhongyun, Zhang, Yinxing, Bao, Han, Huang, Hejiao, Zhou, Yicong
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Cats; Channel noise; Chaos theory; Chaotic communication; Chaotic system; Complexity theory; Degradation; Dynamical systems; Eigenvalues and eigenfunctions; Hardware; hardware implementation; Liapunov exponents; Linear algebra; Mathematical analysis; Microcontrollers; nonlinear system; Parameters; Performance evaluation; Polynomials; random number generator; secure communication
• DOI: 10.1109/TCSI.2021.3117865
• ISSN: 1549-8328

Designing high-dimensional chaotic maps with expected dynamic properties is an attractive but challenging task. The dynamic properties of a chaotic system can be reflected by the Lyapunov exponents (LEs). Using the inherent relationship between the parameters of a chaotic map and its LEs, this paper proposes an <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>-dimensional polynomial chaotic system (<inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS) that can generate <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula> chaotic maps with any desired LEs. The <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS is constructed from <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> parametric polynomials with arbitrary orders, and its parameter matrix is configured using the preliminaries in linear algebra. Theoretical analysis proves that the <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS can produce high-dimensional chaotic maps with any desired LEs. To show the effects of the <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS, two high-dimensional chaotic maps with hyperchaotic behaviors were generated. A microcontroller-based hardware platform was developed to implement the two chaotic maps, and the test results demonstrated the randomness properties of their chaotic signals. Performance evaluations indicate that the high-dimensional chaotic maps generated from <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS have the desired LEs and more complicated dynamic behaviors compared with other high-dimensional chaotic maps. In addition, to demonstrate the applications of <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS, we developed a chaos-based secure communication scheme. Simulation results show that <inline-formula> <tex-math notation="LaTeX">n\text{D} </tex-math></inline-formula>-PCS has a stronger ability to resist channel noise than other high-dimensional chaotic maps.