Application of Caputo-Fabrizio derivative in circuit realization

A. M. Alqahtani; Shivani Sharma; Arun Chaudhary; Aditya Sharma; A. M. Alqahtani; Shivani Sharma; Arun Chaudhary; Aditya Sharma

doi:10.3934/math.2025113

AIMS Mathematics

2025, Volume 10, Issue 2: 2415-2443. doi: 10.3934/math.2025113

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Research article

Application of Caputo-Fabrizio derivative in circuit realization

1.
Department of Mathematics, Shaqra University, Riyadh, Saudi Arabia
2.
Department of Mathematics, Amity University Rajasthan, Jaipur, India
3.
Department of Mathematics, Rajdhani College, University of Delhi, Delhi, India
4.
Department of Electronics Science, University of Delhi, Delhi, India

Received: 17 November 2024 Revised: 06 December 2024 Accepted: 09 December 2024 Published: 11 February 2025
MSC : 34H10, 34C28, 37M05, 26A33

This work incorporates a hyperchaotic system with fractional order Caputo-Fabrizio derivative. A comprehensive proof of the existence followed by the uniqueness is detailed and the simulations are performed at distinct fractional orders. Our results are strengthened by bridging a gap between the theoretical problems and real-world applications. For this, we have implemented a 4D fractional order Caputo-Fabrizio hyperchaotic system to circuit equations. The chaotic behaviors obtained through circuit realization are compared with the phase portrait diagrams obtained by simulating the fractional order Caputo-Fabrizio hyperchaotic system. In this work, we establish a foundation for future advancements in circuit integration and the development of complex control systems. Subsequently, we can enhance our comprehension of fractional order hyperchaotic systems by incorporating with practical experiments.
- fractional derivative,
- hyperchaotic system,
- numerical solutions,
- simulations,
- nonlinear dynamical systems,
- circuit realization
Citation: A. M. Alqahtani, Shivani Sharma, Arun Chaudhary, Aditya Sharma. Application of Caputo-Fabrizio derivative in circuit realization[J]. AIMS Mathematics, 2025, 10(2): 2415-2443. doi: 10.3934/math.2025113

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Abstract

This work incorporates a hyperchaotic system with fractional order Caputo-Fabrizio derivative. A comprehensive proof of the existence followed by the uniqueness is detailed and the simulations are performed at distinct fractional orders. Our results are strengthened by bridging a gap between the theoretical problems and real-world applications. For this, we have implemented a 4D fractional order Caputo-Fabrizio hyperchaotic system to circuit equations. The chaotic behaviors obtained through circuit realization are compared with the phase portrait diagrams obtained by simulating the fractional order Caputo-Fabrizio hyperchaotic system. In this work, we establish a foundation for future advancements in circuit integration and the development of complex control systems. Subsequently, we can enhance our comprehension of fractional order hyperchaotic systems by incorporating with practical experiments.

References

[1]	K. B. Oldham, J. Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, Mathematics in Science and Engineering, Academic Press, 111 (1974).
[2]	M. Caputo, M. Fabrizio, A new deﬁnition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[3]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[4]	A. B. M. Alzahrani, R. Saadeh, M. A. Abdoon, M. Elbadri, M. Berir, A. Qazza, Effective methods for numerical analysis of the simplest chaotic circuit model with Atangana-Baleanu Caputo fractional derivative, J. Eng. Math., 144 (2024), 9. https://doi.org/10.1007/s10665-023-10319-x doi: 10.1007/s10665-023-10319-x
[5]	L. Song, W. Yu, Y. Tan, K. Duan, Calculations of fractional derivative option pricing models based on neural network, J. Comput. Appl. Math., 437 (2024), 115462. https://doi.org/10.1016/j.cam.2023.115462 doi: 10.1016/j.cam.2023.115462
[6]	A. M. Alqahtani, A. Chaudhary, R. S. Dubey, S. Sharma, Comparative analysis of the chaotic behavior of a five-dimensional fractional hyperchaotic system with constant and variable order, Fractal Fract., 8 (2024), 421. https://doi.org/10.3390/fractalfract8070421 doi: 10.3390/fractalfract8070421
[7]	A. D. Matteo, P. D. Spanos, Determination of nonstationary stochastic response of linear oscillators with fractional derivative elements of rational Order, ASME. J. Appl. Mech., 91 (2024), 041008. https://doi.org/10.1115/1.4064143 doi: 10.1115/1.4064143
[8]	G. Barbero, L. R. Evangelista, R. S. Zola, E. K. Lenzi, A. M. Scarfone, A brief review of fractional calculus as a tool for applications in physics: Adsorption phenomena and electrical impedance in complex fluids, Fractal Fract., 8 (2024), 369. https://doi.org/10.3390/fractalfract8070369 doi: 10.3390/fractalfract8070369
[9]	R. Wang, Y. Sui, Method for measuring the fractional derivative of a two-dimensional magnetic field based on Taylor-Riemann series, Fractal Fract., 8 (2024), 375. https://doi.org/10.3390/fractalfract8070375 doi: 10.3390/fractalfract8070375
[10]	S. Sharma, R. S. Dubey, A. Chaudhary, Caputo fractional model for the predator-prey relation with sickness in prey and refuge to susceptible prey, Int. J. Math. Ind., 1 (2024), 2450014. https://doi.org/10.1142/S266133522450014X doi: 10.1142/S266133522450014X
[11]	M. A. Almuqrin, P. Goswami, S. Sharma, I. Khan, R. S. Dubey, A. Khan, Fractional model of Ebola virus in population of bats in frame of Atangana-Baleanu fractional derivative, Results Phys., 26 (2021), 104295. https://doi.org/10.1016/j.rinp.2021.104295 doi: 10.1016/j.rinp.2021.104295
[12]	S. Althubiti, S. Sharma, P. Goswami, R. S. Dubey, Fractional SIAQR model with time dependent infection rate, Arab J. Basic Appl. Sci., 30 (2023), 307–316. https://doi.org/10.1080/25765299.2023.2209981 doi: 10.1080/25765299.2023.2209981
[13]	B. B. Mandelbrot, Fractal geometry: What is it, and what does it do? Proc. R. Soc. Lond., 423 (1989), 3–16. https://doi.org/10.1098/rspa.1989.0038 doi: 10.1098/rspa.1989.0038
[14]	J. H. He, A new fractal derivation, Therm. Sci., 15 (2011), 145–147. https://doi.org/10.2298/TSCI11S1145H doi: 10.2298/TSCI11S1145H
[15]	J. H. He, Y. O. El-DiB, A tutorial introduction to the two-scale fractal calculus and its application to the fractal Zhiber-Shabat Oscillator, Fractals, 29 (2021), 2150268. https://doi.org/10.1142/S0218348X21502686 doi: 10.1142/S0218348X21502686
[16]	C. H. He, H. Liu, C. Liu, A fractal-based approach to the mechanical properties of recycled aggregate concretes, Facta. Univ.-Ser. Mech., 22 (2024), 329–342. https://doi.org/10.22190/fume240605035h doi: 10.22190/fume240605035h
[17]	C. H. He, C. Liu, Fractal dimensions of a porous concrete and its effect on the concrete's strength, Facta. Univ.-Ser. Mech., 21 (2023), 137–150. https://doi.org/10.22190/fume221215005h doi: 10.22190/fume221215005h
[18]	S. Abbas, Z. U. Nisa, M. Nazar, M. Amjad, H. Ali, A. Z. Jan, Application of heat and mass transfer to convective flow of Casson fluids in a microchannel with Caputo-Fabrizio derivative approach, Arab J. Sci. Eng., 49 (2024), 1275–1286. https://doi.org/10.1007/s13369-023-08351-1 doi: 10.1007/s13369-023-08351-1
[19]	B. S. N. Murthy, M. N. Srinivas, V. Madhusudanan, A. Zeb, E. M. T. Eldin, S. Etemad, et al., The impact of Caputo-Fabrizio fractional derivative and the dynamics of noise on worm propagation in wireless IoT networks, Alex. Eng. J., 91 (2024), 558–579. https://doi.org/10.1016/j.aej.2024.02.027 doi: 10.1016/j.aej.2024.02.027
[20]	A. Chavada, N. Pathak, Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data, Math. Model. Control., 4 (2024), 119–132. https://doi.org/10.3934/mmc.2024011 doi: 10.3934/mmc.2024011
[21]	O. E. Rossler, An equation for hyperchaos, Physics Lett. A., 71 (1979), 155–157. https://doi.org/10.1016/0375-9601(79)90150-6 doi: 10.1016/0375-9601(79)90150-6
[22]	U. Erkan, A. Toktas, Q. Lai, 2D hyperchaotic system based on Schaffer function for image encryption, Expert Syst. Appl., 213 (2023), 119076. https://doi.org/10.1016/j.eswa.2022.119076 doi: 10.1016/j.eswa.2022.119076
[23]	S. Zhu, X. Deng, W. Zhang, C. Zhu, Construction of a new 2D hyperchaotic map with application in efficient pseudo-random number generator design and color image encryption, Mathematics., 11 (2023), 3171. https://doi.org/10.3390/math11143171 doi: 10.3390/math11143171
[24]	J. Tang, Z. Zhang, T. Huang, Two-dimensional cosine-sine interleaved chaotic system for secure communication, IEEE T. Circuits-II, 71 (2024), 2479–2483. https://doi.org/10.1109/TCSII.2023.3337145 doi: 10.1109/TCSII.2023.3337145
[25]	W. Li, W. Yan, R. Zhang, C. Wang, Q. Ding, A new 3D discrete hyperchaotic system and its application in secure transmission, Int. J. Bifurc. Chaos Appl., 29 (2019), 1950206. https://doi.org/10.1142/S0218127419502067 doi: 10.1142/S0218127419502067
[26]	W. J. Zhi, C. Z. Qiang, Y. Z. Zhi, The generation of a hyperchaotic system based on a three-dimensional autonomous chaotic system, Chinese Phys., 15 (2006), 6. https://doi.org/10.1088/1009-1963/15/6/015 doi: 10.1088/1009-1963/15/6/015
[27]	A. Sahasrabuddhe, D. S. Laiphrakpam, Multiple images encryption based on 3D scrambling and hyper-chaotic system, Inf. Sci., 550 (2021), 252–267. https://doi.org/10.1016/j.ins.2020.10.031 doi: 10.1016/j.ins.2020.10.031
[28]	H. Wang, G. Dong, New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system, Appl. Math. Comput., 346 (2019), 272–286. https://doi.org/10.1016/j.amc.2018.10.006 doi: 10.1016/j.amc.2018.10.006
[29]	T. Nestor, A. Belazi, B. Abd-El-Atty, M. N. Aslam, C. Volos, N. J. De Dieu, et al., A new 4D hyperchaotic system with dynamics analysis, synchronization, and application to image encryption, Symmetry, 14 (2022), 424. https://doi.org/10.3390/sym14020424 doi: 10.3390/sym14020424
[30]	X. Liu, X. Tong, M. Zhang, Z. Wang, A highly secure image encryption algorithm based on conservative hyperchaotic system and dynamic biogenetic gene algorithms, Chaos Soliton. Fract., 171 (2023), 113450. https://doi.org/10.1016/j.chaos.2023.113450 doi: 10.1016/j.chaos.2023.113450
[31]	Y. Hui, H. Liu, P. Fang, A DNA image encryption based on a new hyperchaotic system, Multimed. Tools Appl., 82 (2023), 21983–22007. https://doi.org/10.1007/s11042-021-10526-7 doi: 10.1007/s11042-021-10526-7
[32]	R. B. Naik, U. Singh, A review on applications of chaotic maps in psuedo-random number generators and encryption, Ann. Data Sci., 11 (2024), 25–50. https://doi.org/10.1007/s40745-021-00364-7 doi: 10.1007/s40745-021-00364-7
[33]	T. Bonny, R. A. Debsi, S. Majzoub, A. S. Elwakil, Hardware optimized FPGA implementations of high-speed true random bit generators based on switching-type chaotic oscillators, Circ. Syst. Signal Pr., 38 (2019), 1342–1359. https://doi.org/10.1007/s00034-018-0905-6 doi: 10.1007/s00034-018-0905-6
[34]	K. W. Wong, Q. Z. Liu, J. Y. Chen, Simultaneous arithmetic coding and encryption using chaotic maps, IEEE T. Circuits-II, 57 (2010), 146–150. https://doi.org/10.1109/TCSII.2010.2040315 doi: 10.1109/TCSII.2010.2040315
[35]	Z. N. A. Kateeb, S. J. Mohammed, Encrypting an audio file based on integer wavelet transform and hand geometry, Telkomnika, 18 (2020). https://doi.org/10.12928/TELKOMNIKA.v18i4.14216 doi: 10.12928/TELKOMNIKA.v18i4.14216
[36]	K. Benkouider, A. Sambas, T. Bonny, W. A. Nassan, I. A. R. Moghrabi, I. M. Sulaiman, et al., A comprehensive study of the novel 4D hyperchaotic system with self-exited multistability and application in the voice encryption, Sci. Rep., 14 (2024), 12993. https://doi.org/10.1038/s41598-024-63779-1 doi: 10.1038/s41598-024-63779-1
[37]	H. Wen, Y. Lin, Z. Xie, T. Liu, Chaos-based block permutation and dynamic sequence multiplexing for video encryption, Sci. Rep., 13 (2023), 14721. https://doi.org/10.1038/s41598-023-41082-9 doi: 10.1038/s41598-023-41082-9
[38]	A. Yousefpour, H. Jahanshahi, J. M. M. Pacheco, S. Bekiros, Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos, Chaos Soliton. Fract., 130 (2020), 109400. https://doi.org/10.1016/j.chaos.2019.109400 doi: 10.1016/j.chaos.2019.109400
[39]	L. Pan, W. Zhou, L. Zhou, K. Sun, Chaos synchronization between two different fractional-order hyperchaotic systems, Commun. Nonlinear Sci., 16 (2011), 2628–2640. https://doi.org/10.1016/j.cnsns.2010.09.016 doi: 10.1016/j.cnsns.2010.09.016
[40]	Q. Shi, X. An, L. Xiong, F. Yang, L. Zhang, Dynamic analysis of a fractional-order hyperchaotic system and its application in image encryption, Phys. Scr., 97 (2022), 045201. https://doi.org/10.1088/1402-4896/ac55bb doi: 10.1088/1402-4896/ac55bb
[41]	H. Deng, T. Li, Q. Wang, H. Li, A fractional-order hyperchaotic system and its synchronization, Chaos Soliton. Fract., 41 (2009), 962–969. https://doi.org/10.1016/j.chaos.2008.04.034 doi: 10.1016/j.chaos.2008.04.034
[42]	M. A. Balootaki, H. Rahmani, H. Moeinkhah, A. Mohammadzadeh, On the synchronization and stabilization of fractional-order chaotic systems: Recent advances and future perspectives, Physica A, 551 (2020), 124203. https://doi.org/10.1016/j.physa.2020.124203 doi: 10.1016/j.physa.2020.124203
[43]	Z. A. S. A. Rahman, B. H. Jasim, Y. I. A. Al-Yasir, Y. F. Hu, R. A. A. Alhameed, B. N. Alhasnawi, A new fractional-order chaotic system with its analysis, synchronization, and circuit realization for secure communication applications, Mathematics, 9 (2021), 2593. https://doi.org/10.3390/math9202593 doi: 10.3390/math9202593
[44]	O. M. Fuentes, J. J. M. Garcia, J. F. G. Aguilar, Generalized synchronization of commensurate fractional-order chaotic systems: Applications in secure information transmission. Digit. Signal Process., 126 (2022), 103494. https://doi.org/10.1016/j.dsp.2022.103494 doi: 10.1016/j.dsp.2022.103494
[45]	S. Xu, X. Wang, X. Ye, A new fractional-order chaos system of Hopfield neural network and its application in image processing, Chaos Soliton. Fract., 157 (2022), 111889. https://doi.org/10.1016/j.chaos.2022.111889 doi: 10.1016/j.chaos.2022.111889
[46]	L. Chen, H. Yin, T. Huang, L. Yuan, S. Zheng, L. Yin, Chaos in fractional-order discrete neural networks with application to image encryption, Neural Networks, 125 (2020), 174–184. https://doi.org/10.1016/j.neunet.2020.02.008 doi: 10.1016/j.neunet.2020.02.008
[47]	S. Fu, X. F. Cheng, J. Liu, Dynamics, circuit design, feedback control of a new hyperchaotic system and its application in audio encryption, Sci. Rep., 13 (2023), 19385. https://doi.org/10.1038/s41598-023-46161-5 doi: 10.1038/s41598-023-46161-5
[48]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92. https://doi.org/10.12785/pfda/010202 doi: 10.12785/pfda/010202
[49]	M. Toufik, A. Atangana, New numerical approximation of fractional derivative with non-local and non-singular kernel, Eur. Phys. J. Plus., 132 (2017), 444. https://doi.org/10.1140/epjp/i2017-11717-0 doi: 10.1140/epjp/i2017-11717-0
[50]	R. Batogna, A. Atangana, New two step Laplace Adam-Bashforth method for integer and noninteger order partial differential equations, Numer. Meth. Part. D. E., 34 (2017), 1739–1785. https://doi.org/10.1002/num.22216 doi: 10.1002/num.22216

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