Optimizing chaotic systems by orbit counting and Fourier spectrum: FPGA implementation and image encryption application
Abstract
The optimization of chaotic systems has been performed by considering dynamical characteristics of the mathematical models. The proposed work shows the application of genetic algorithms (GAs) to optimize the chaotic behavior of three well-known systems, namely: Lorenz, Chen and Lü. The parameters of the chaotic systems are varied in a specific range of values considered as the search space, and the evaluation of the mathematical model is performed by applying the Forward Euler method. The contribution presented herein is that the chaotic behavior is evaluated by counting the orbits in an attractor and the sparsity of them. In addition, the chaotic behavior is guaranteed by evaluating the Fourier spectrum of the time series. The solutions provided by the GA, are then implemented on a field-programmable gate array (FPGA) to verify the experimental generation of chaotic attractors. Finally, two optimized chaotic systems are synchronized and used to encrypt an image, thus confirming the appropriateness of optimizing the chaotic behavior by orbit counting and Fourier spectrum analysis.
Keyword : chaotic system, genetic algorithm, orbit, Fourier spectrum, FPGA, image encryption
How to Cite
Rojas-Galván, R., Mercado-Gutiérrez, E., & Tlelo-Cuautle, E. (2025). Optimizing chaotic systems by orbit counting and Fourier spectrum: FPGA implementation and image encryption application. Mathematical Modelling and Analysis, 30(2), 322–341. https://doi.org/10.3846/mma.2025.22089
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Dai and L.-H. Fu. Research on the hybrid chaos-coud salp swarm algorithm. Commun. Nonlinear Sci. Numer. Simul., 138, Nov. 2024.
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M. Devipriya and M. Brindha. Reconfigurable architecture for DNA diffusion technique-based medical image encryption. JCSC, 32(4), Mar. 15 2023. https://doi.org/10.1142/S0218126623500652
M. Devipriya, M. Sreenivasan and M. Brindha. Reconfigurable architecture for image encryption using a three-layer artificial neural network. IETE Journal of Research, 70(1):473–486, Jan. 2 2024. https://doi.org/10.1080/03772063.2022.2127940
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M. Jornet. Generalized polynomial chaos expansions for the random fractional bateman equations. Appl. Math. Comput., 479, Oct. 15 2024. https://doi.org/10.1016/j.amc.2024.128873
H. Khujamatov, M. Pitchai, A. Shamsiev, A. Mukhamadiyev and J. Cho. Clustered routing using chaotic genetic algorithm with grey wolf optimization to enhance energy efficiency in sensor networks. Sensors, 24(13), Jul. 2024. https://doi.org/10.3390/s24134406
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J. Lü and G. Chen. A new chaotic attractor coined. IJBC, 12(03):659–661, 2002. https://doi.org/10.1142/S0218127402004620
R. Malhotra, N. Singh and Y. Singh. Genetic algorithms: Concepts, design for optimization of process controllers. Computer and Information Science, 4(2):39– 54, 2011. https://doi.org/10.5539/cis.v4n2p39
M.M. Miah, F. Alsharif, Md.A. Iqbal, J.R.M. Borhan and M. Kanan. Chaotic phenomena, sensitivity analysis, bifurcation analysis, and new abundant solitary wave structures of the two nonlinear dynamical models in industrial optimization. Math., 12(13), Jul. 2024. https://doi.org/10.3390/math12131959
D. Ogorelova and F. Sadyrbaev. Comparative analysis of models of genetic and neuronal networks. Math. Model. Anal., 29(2):277–287, 2024. https://doi.org/10.3846/mma.2024.19714
M. F. Parra-Ocampo, O. Serrano-Perez, A. Rodriguez-Molina, M.G. VillarrealCervantes, G. Hernandez, M.E. Sanchez-Gutierrez and V.M. Silva-Garcia. Enhancing the performance in the offline controller tuning of robotic manipulators with chaos: a comparative study with differential evolution. Int. J. Dynam. Control, 2024 Apr. 11 2024. https://doi.org/10.1007/s40435-024-01423-6
L.M. Pecora and T.L. Carroll. Synchronization in chaotic systems. Phys. Rev. Lett., 64(8):821–824, Feb. 19 1990. https://doi.org/10.1103/PhysRevLett.64.821
H.M. Perez and J. Solis-Daun. Global stabilization of a bounded controlled lorenz system. IJBC, 34(07), Jun. 15 2024. https://doi.org/10.1142/S0218127424500895
P. Pitchandi, M. Nivaashini and R.K. Grace. Optimized ensemble learning models based on clustering and hybrid deep learning for wireless intrusion detection. IETE Journal of Research, 2024 Jun. 19 2024. https://doi.org/10.1080/03772063.2024.2367042
N.P. Ponnuviji, E. Nirmala, F.M. H. Fernandez and K. Anitha. ALRN-RCS: Advanced approach to network intrusion detection using attention long-term recurrent networks and chaotic optimization. IETE Journal of Research, 2024 Jul. 24 2024. https://doi.org/10.1080/03772063.2024.2370949
X. Qin and Y. Zhang. Image encryption algorithm based on coa and hyperchaotic lorenz system. Nonlinear Dyn., 112(12):10611–10632, Jun. 2024. https://doi.org/10.1007/s11071-024-09632-6
M. Tadj, L. Chaib, A. Choucha, M. Alhazmi, A. Alwabli, M. Bajaj and S.A. Dost Mohammadi. Improved chaotic bat algorithm for optimal coordinated tuning of power system stabilizers for multimachine power system. Sci. Rep., 14(1), Jul. 2 2024. https://doi.org/10.1038/s41598-024-65101-5
E. Tlelo-Cuautle, L.G. De La Fraga, O. Guillén-Fernández and A. SilvaJuárez. Optimization of Integer/Fractional Order Chaotic Systems by Metaheuristics and Their Electronic Realization. CRC press, 2021. https://doi.org/10.1201/9781003042181
A. Toktas, U. Erkan, D. Ustun and Q. Lai. Multiobjective design of 2d hyperchaotic system using leader pareto grey wolf optimizer. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2024 Jun. 6 2024. https://doi.org/10.1109/TSMC.2024.3401412
A. Wadood, B.S. Khan, H. Albalawi and A.M. Alatwi. Design of the novel fractional order hybrid whale optimizer for thermal wind power generation systems with integration of chaos infused wind power. Fractal. Fract., 8(7), Jul. 2024. https://doi.org/10.3390/fractalfract8070379
D.S. Weile and E. Michielssen. Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering and System Safety, 91(9):992–1007, 2006. https://doi.org/10.1016/j.ress.2005.11.018
F. Yu, S. Xu, Y. Lin, Y.M. Gracia, W. Yao and Sh. Cai. Dynamic analysis, image encryption application and fpga implementation of a discrete memristor-coupled neural network. IJBC, 34(06), May. 2024. https://doi.org/10.1142/S0218127424500688
F. Yu, Sh. Xu, X. Xiao, W. Yao, Y. Huang, Sh. Cai and Y. Li. Dynamic analysis and fpga implementation of a 5d multi-wing fractional-order memristive chaotic system with hidden attractors. Integration, the VLSI Journal, 96, May. 2024. https://doi.org/10.1016/j.vlsi.2023.102129
J. Yu, W. Xie and L. Zhang. Multiobjective optimization of chaotic image encryption based on ABC algorithm and DNA coding. IJBC, 34(05), Apr. 2024. https://doi.org/10.1142/S0218127424500573
J. Zhang, J. Yang, L. Xu and X. Zhu. The circuit realization of a fifth-order multi-wing chaotic system and its application in image encryption. Int. J. Circuit Theory Appl., 51(3):1168–1186, Mar. 2023. https://doi.org/10.1002/cta.3490