A two-dimensional hyperchaotic discrete dynamical system and its application

Min Zhao; Min Zhao

doi:10.3934/math.2026339

AIMS Mathematics

2026, Volume 11, Issue 3: 8242-8270. doi: 10.3934/math.2026339

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Research article

A two-dimensional hyperchaotic discrete dynamical system and its application

Min Zhao ^,

School of Mathematics, Nanjing University, Nanjing 210093, Jiangsu, China

Received: 01 February 2026 Revised: 09 March 2026 Accepted: 18 March 2026 Published: 27 March 2026
MSC : 37D45, 37M10, 68P25, 94A60

This paper develops a two-dimensional hyperchaotic discrete dynamical system (2D-HDDS) by extending the exponential-Chebyshev mechanism to a two-dimensional setting through a structured nonlinear feedback coupling. The resulting system combines the stretching effect of exponential dynamics with the folding behavior of Chebyshev mappings. Bifurcation diagrams and Lyapunov-exponent spectra indicate that 2D-HDDS exhibits a comparatively wide hyperchaotic parameter region and high dynamical complexity. Based on the generated keystream, we further construct an image-encryption scheme that integrates dual permutations, a key-dependent dynamic S-box, index-coupled forward and backward nonlinear diffusion, and a lightweight avalanche booster to enhance the propagation of plaintext perturbations. Experimental results show high key sensitivity and strong plaintext sensitivity, with the number of pixels change rate (NPCR) and the unified average changing intensity (UACI) values close to the ideal levels predicted by the standard random-like cipher-image model, while also demonstrating good resistance to statistical and differential attacks.
- hyperchaotic system,
- Lyapunov exponent,
- image encryption,
- S-box,
- security analysis
Citation: Min Zhao. A two-dimensional hyperchaotic discrete dynamical system and its application[J]. AIMS Mathematics, 2026, 11(3): 8242-8270. doi: 10.3934/math.2026339

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Abstract

This paper develops a two-dimensional hyperchaotic discrete dynamical system (2D-HDDS) by extending the exponential-Chebyshev mechanism to a two-dimensional setting through a structured nonlinear feedback coupling. The resulting system combines the stretching effect of exponential dynamics with the folding behavior of Chebyshev mappings. Bifurcation diagrams and Lyapunov-exponent spectra indicate that 2D-HDDS exhibits a comparatively wide hyperchaotic parameter region and high dynamical complexity. Based on the generated keystream, we further construct an image-encryption scheme that integrates dual permutations, a key-dependent dynamic S-box, index-coupled forward and backward nonlinear diffusion, and a lightweight avalanche booster to enhance the propagation of plaintext perturbations. Experimental results show high key sensitivity and strong plaintext sensitivity, with the number of pixels change rate (NPCR) and the unified average changing intensity (UACI) values close to the ideal levels predicted by the standard random-like cipher-image model, while also demonstrating good resistance to statistical and differential attacks.

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