https://doi.org/10.1140/epjp/s13360-026-07699-1
Regular Article
Dynamical behavior of a four-dimensional discrete fractional-order polynomial chaotic map and its applications: image encryption and FPGA implementation
1
College of Science, Northwest A&F University, 712100, Yangling, Shaanxi, China
2
College of Mechanical and Electronic Engineering, Northwest A&F University, 712100, Yangling, Shaanxi, China
3
Department of Computing, The Hong Kong Polytechnic University, 999077, Hong Kong, China
a
This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
4
March
2026
Accepted:
15
April
2026
Published online:
3
May
2026
Abstract
A detailed analysis encompassing chaotic trajectories, Lyapunov exponents, and equilibrium points is conducted to investigate the dynamic characteristics and identify hidden attractors of the proposed four-dimensional discrete fractional-order polynomial chaotic map. The implementation of discrete fractional-order chaotic systems with memory effects on hardware with limited resources is intriguing but challenging. Conventional engineering approaches tend to implement such systems in the frequency domain using Z-transform. However, this study proposes an alternative time-domain FPGA implementation based on the theory of short-memory fractional difference equations. The proposed implementation successfully generates chaotic sequences, with their accuracy verified through numerical simulations. In addition, we develop a novel chaotic synchronization control law based on stability criteria for fractional difference equations. The system is further extended to implement chaos-based image encryption. The experimental results demonstrate that the FPGA-implemented chaotic encryption system achieves satisfactory performance with a considerably large key space. It is noteworthy that the two synchronized systems can be utilized separately for image encryption and decryption processes, respectively, thereby enhancing the algorithm’s security. In practice, short-memory difference equations offer computational advantages by reducing memory requirements.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
