FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION

Gülnur Yılmaz; Enis Günay

doi:10.17482/uumfd.1339620

Research Article

4-Boyutlu Hiperkaotik Pang Sisteminin Kesir Dereceli Analizi ve Adaptif Senkronizasyonu

Year 2024, Volume: 29 Issue: 1, 85 - 100, 22.04.2024

Gülnur Yılmaz Enis Günay

https://doi.org/10.17482/uumfd.1339620

Abstract

Kesir dereceli hesaplamalar doğrusal olmayan sistemlerin dinamiklerini analiz etmekte kullanılan ve daha kesin sonuçlar elde edilmesini sağlayan etkili bir yöntemdir. Bu çalışmada, öncelikle 4 boyutlu Pang sistemi tanıtılmış ve hiperkaotik yapısını gösteren dinamik analizleri verilmiştir. Daha sonra sistemin kesir dereceli hesaplamaları yapılarak farklı kesir dereceleri için sahip olduğu dinamikler incelenmiştir. Bu kapsamda, Lyapunov üstelleri ve faz-uzayı gösteriminden elde edilen sonuçlara göre, Pang sistemi farklı kesir derecelerinde periyodik, kaotik ve hiperkaotik davranışlar sergilemektedir. Çalışmanın sonunda elde edilen sonuçlar, sistemin 3,52 kesir derecesi için hiperkaotik yapıda olduğunu göstermiştir. Elde edilen bu sonuç, tamsayı dereceli modele göre kesir dereceli yapı ile daha kesin sonuçlara ulaşıldığını doğrulamıştır. Çalışmanın ilerleyen kısmında, elde edilen kesir dereceli sistemin adaptif senkronizasyonu gerçekleştirilmiştir. Üç farklı durum incelenerek her durumda senkronizasyonun sağlandığı gösterilmiştir.

Keywords

Kesir dereceli sistemler, Senkronizasyon, Hiperkaos

References

1. Abd El-Maksoud, A. J., Abd El-Kader, A. A., Hassan, B. G., Rihan, N. G., Tolba, M. F., Said, L. A., Radwan, A. G., & Abu-Elyazeed, M. F. (2019). FPGA implementation of sound encryption system based on fractional-order chaotic systems. Microelectronics Journal, 90, 323–335. https://doi.org/10.1016/j.mejo.2019.05.005
2. Al-Obeidi, A. S., & AL-Azzawi, S. F. (2019). Projective synchronization for a cass of 6-D hyperchaotic lorenz system. Indonesian Journal of Electrical Engineering and Computer Science, 16(2), 692– 700. https://doi.org/10.11591/IJEECS.V16.I2.PP692-700
3. Bouridah, M. S., Bouden, T., & Yalçin, M. E. (2021). Delayed outputs fractional-order hyperchaotic systems synchronization for images encryption. Multimedia Tools and Applications 2021 80:10, 80(10), 14723–14752. https://doi.org/10.1007/S11042-020-10425-3
4. Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent—II. Geophysical Journal International, 13(5), 529–539. https://doi.org/10.1111/J.1365-246X.1967.TB02303.X
5. Gularte, K. H. M., Alves, L. M., Vargas, J. A. R., Alfaro, S. C. A., De Carvalho, G. C., & Romero, J. F. A. (2021). Secure Communication Based on Hyperchaotic Underactuated Projective Synchronization. IEEE Access, 9, 166117–166128. https://doi.org/10.1109/ACCESS.2021.3134829
6. Huang, W., Jiang, D., An, Y., Liu, L., & Wang, X. (2021). A Novel Double-Image Encryption Algorithm Based on Rossler Hyperchaotic System and Compressive Sensing. IEEE Access, 9, 41704–41716. https://doi.org/10.1109/ACCESS.2021.3065453
7. Liao, T. L., Wan, P. Y., & Yan, J. J. (2022). Design and synchronization of chaos-based true random number generators and its FPGA implementation. IEEE Access. https://doi.org/10.1109/ACCESS.2022.3142536
8. Lin, L., Wang, Q., & Cai, G. (2022). FPGA Realization of Two Different Fractional- Order Time-Delay Chaotic System With Predefined Synchronization Time. IEEE Access, 10, 133663–133672. https://doi.org/10.1109/ACCESS.2022.3231610
9. Lin, L., Wang, Q., He, B., Chen, Y., Peng, X., & Mei, R. (2021). Adaptive predefined-time synchronization of two different fractional-order chaotic systems with time-delay. IEEE Access, 9, 31908–31920. https://doi.org/10.1109/ACCESS.2021.3059324
10. Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20(2), 130–141. https://doi.org/10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2
11. Lu, J. G. (2006). Chaotic dynamics of the fractional-order Lü system and its synchronization. Physics Letters A, 354(4), 305–311. https://doi.org/10.1016/J.PHYSLETA.2006.01.068
12. Meng, X., Wu, Z., Gao, C., Jiang, B., & Karimi, H. R. (2021). Finite-time projective synchronization control of variable-order fractional chaotic systems via sliding mode approach. IEEE Transactions on Circuits and Systems II: Express Briefs, 68(7), 2503–2507. https://doi.org/10.1109/TCSII.2021.3055753
13. Nwachioma, C., & Pérez-Cruz, J. H. (2021). Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot. Chaos, Solitons and Fractals, 144, 110684. https://doi.org/10.1016/J.CHAOS.2021.110684
14. Oldham, K. B., & Spanier, J. (1974). The fractional calculus : theory and applications of differentiation and integration to arbitrary order. 234.
15. Pang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775–2789. https://doi.org/10.1016/j.cam.2010.11.029
16. Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar, D. J., & Heagy, J. F. (1997). Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos, 7(4), 520–543. https://doi.org/10.1063/1.166278
17. Qammer, H. K. (1995). Chaos in a Fractional Order Chua’s System. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42(8), 485–490. https://doi.org/10.1109/81.404062
18. Sajjadi, S. S., Baleanu, D., Jajarmi, A., & Pirouz, H. M. (2020). A new adaptive synchronization and hyperchaos control of a biological snap oscillator. Chaos, Solitons and Fractals, 138. https://doi.org/10.1016/j.chaos.2020.109919
19. Scherer, R., Kalla, S. L., Tang, Y., & Huang, J. (2011). The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications, 62(3), 902–917. https://doi.org/10.1016/J.CAMWA.2011.03.054
20. Singh, S., Han, S., & Lee, S. M. (2021). Adaptive single input sliding mode control for hybrid-synchronization of uncertain hyperchaotic Lu systems. Journal of the Franklin Institute. https://doi.org/10.1016/J.JFRANKLIN.2021.07.037
21. Vaidyanathan, S., Sambas, A., Mujiarto, Mamat, M., Wilarso, Mada Sanjaya, W. S., Sutoni, A., & Gunawan, I. (2021). A New 4-D Multistable Hyperchaotic Two-Scroll System, its Bifurcation Analysis, Synchronization and Circuit Simulation. Journal of Physics: Conference Series, 1764(1). https://doi.org/10.1088/1742-6596/1764/1/012206
22. Wang, F., Wang, R., Iu, H. H. C., Liu, C., & Fernando, T. (2019). A Novel Multi-Shape Chaotic Attractor and Its FPGA Implementation. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(12), 2062–2066. https://doi.org/10.1109/TCSII.2019.2907709
23. Wang, J., Yu, W., Wang, J., Zhao, Y., Zhang, J., & Jiang, D. (2019). A new six-dimensional hyperchaotic system and its secure communication circuit implementation. International Journal of Circuit Theory and Applications, 47(5), 702–717. https://doi.org/10.1002/CTA.2617
24. Wang, P., Wen, G., Yu, X., Yu, W., & Huang, T. (2019). Synchronization of multi-layer networks: From node-to-node synchronization to complete synchronization. IEEE Transactions on Circuits and Systems I: Regular Papers, 66(3), 1141–1152. https://doi.org/10.1109/TCSI.2018.2877414
25. Wang, S., Hong, L., Jiang, J., & Li, X. (2020). Synchronization precision analysis of a fractional-order hyperchaos with application to image encryption. AIP Advances, 10(10). https://doi.org/10.1063/5.0012493
26. Wu, X., Lu, H., & Shen, S. (2009). Synchronization of a new fractional-order hyperchaotic system. Physics Letters, Section A: General, Atomic and Solid State Physics, 373(27–28), 2329–2337. https://doi.org/10.1016/j.physleta.2009.04.063
27. Yılmaz, G., Altun, K., & Günay, E. (2022). Synchronization of hyperchaotic Wang-Liu system with experimental implementation on FPAA and FPGA. Analog Integrated Circuits and Signal Processing, 113(2), 145–161. https://doi.org/10.1007/S10470-022-02073-4

FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION

Year 2024, Volume: 29 Issue: 1, 85 - 100, 22.04.2024

Gülnur Yılmaz Enis Günay

https://doi.org/10.17482/uumfd.1339620

Abstract

Fractional calculus is an effective method used to analyze the dynamics of nonlinear systems and provide more precise results. In this study, firstly, the 4-dimensional Pang system is introduced and its dynamic analyses demonstrating the hyperchaotic structure are given. Then, fractional-order calculations of the system are presented and the dynamics of the system for different fraction orders are investigated. At this point, according to the results obtained from Lyapunov exponents and phase-space representation, the Pang system exhibits periodic, chaotic, and hyperchaotic behaviors in different fractional orders. The results obtained at the end of this study present that the system is hyperchaotic for the fractional order of 3.52 and it is also confirmed that more accurate results are obtained than the integer-order analysis. In the next part of the study, adaptive synchronization of the fractional-order system is performed. Three different cases are examined and it is demonstrated that synchronization is achieved in all cases.

Keywords

Fractional order systems, Synchronization, Hyperchaos

References

1. Abd El-Maksoud, A. J., Abd El-Kader, A. A., Hassan, B. G., Rihan, N. G., Tolba, M. F., Said, L. A., Radwan, A. G., & Abu-Elyazeed, M. F. (2019). FPGA implementation of sound encryption system based on fractional-order chaotic systems. Microelectronics Journal, 90, 323–335. https://doi.org/10.1016/j.mejo.2019.05.005
2. Al-Obeidi, A. S., & AL-Azzawi, S. F. (2019). Projective synchronization for a cass of 6-D hyperchaotic lorenz system. Indonesian Journal of Electrical Engineering and Computer Science, 16(2), 692– 700. https://doi.org/10.11591/IJEECS.V16.I2.PP692-700
3. Bouridah, M. S., Bouden, T., & Yalçin, M. E. (2021). Delayed outputs fractional-order hyperchaotic systems synchronization for images encryption. Multimedia Tools and Applications 2021 80:10, 80(10), 14723–14752. https://doi.org/10.1007/S11042-020-10425-3
4. Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent—II. Geophysical Journal International, 13(5), 529–539. https://doi.org/10.1111/J.1365-246X.1967.TB02303.X
5. Gularte, K. H. M., Alves, L. M., Vargas, J. A. R., Alfaro, S. C. A., De Carvalho, G. C., & Romero, J. F. A. (2021). Secure Communication Based on Hyperchaotic Underactuated Projective Synchronization. IEEE Access, 9, 166117–166128. https://doi.org/10.1109/ACCESS.2021.3134829
6. Huang, W., Jiang, D., An, Y., Liu, L., & Wang, X. (2021). A Novel Double-Image Encryption Algorithm Based on Rossler Hyperchaotic System and Compressive Sensing. IEEE Access, 9, 41704–41716. https://doi.org/10.1109/ACCESS.2021.3065453
7. Liao, T. L., Wan, P. Y., & Yan, J. J. (2022). Design and synchronization of chaos-based true random number generators and its FPGA implementation. IEEE Access. https://doi.org/10.1109/ACCESS.2022.3142536
8. Lin, L., Wang, Q., & Cai, G. (2022). FPGA Realization of Two Different Fractional- Order Time-Delay Chaotic System With Predefined Synchronization Time. IEEE Access, 10, 133663–133672. https://doi.org/10.1109/ACCESS.2022.3231610
9. Lin, L., Wang, Q., He, B., Chen, Y., Peng, X., & Mei, R. (2021). Adaptive predefined-time synchronization of two different fractional-order chaotic systems with time-delay. IEEE Access, 9, 31908–31920. https://doi.org/10.1109/ACCESS.2021.3059324
10. Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20(2), 130–141. https://doi.org/10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2
11. Lu, J. G. (2006). Chaotic dynamics of the fractional-order Lü system and its synchronization. Physics Letters A, 354(4), 305–311. https://doi.org/10.1016/J.PHYSLETA.2006.01.068
12. Meng, X., Wu, Z., Gao, C., Jiang, B., & Karimi, H. R. (2021). Finite-time projective synchronization control of variable-order fractional chaotic systems via sliding mode approach. IEEE Transactions on Circuits and Systems II: Express Briefs, 68(7), 2503–2507. https://doi.org/10.1109/TCSII.2021.3055753
13. Nwachioma, C., & Pérez-Cruz, J. H. (2021). Analysis of a new chaotic system, electronic realization and use in navigation of differential drive mobile robot. Chaos, Solitons and Fractals, 144, 110684. https://doi.org/10.1016/J.CHAOS.2021.110684
14. Oldham, K. B., & Spanier, J. (1974). The fractional calculus : theory and applications of differentiation and integration to arbitrary order. 234.
15. Pang, S., & Liu, Y. (2011). A new hyperchaotic system from the Lü system and its control. Journal of Computational and Applied Mathematics, 235(8), 2775–2789. https://doi.org/10.1016/j.cam.2010.11.029
16. Pecora, L. M., Carroll, T. L., Johnson, G. A., Mar, D. J., & Heagy, J. F. (1997). Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos, 7(4), 520–543. https://doi.org/10.1063/1.166278
17. Qammer, H. K. (1995). Chaos in a Fractional Order Chua’s System. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42(8), 485–490. https://doi.org/10.1109/81.404062
18. Sajjadi, S. S., Baleanu, D., Jajarmi, A., & Pirouz, H. M. (2020). A new adaptive synchronization and hyperchaos control of a biological snap oscillator. Chaos, Solitons and Fractals, 138. https://doi.org/10.1016/j.chaos.2020.109919
19. Scherer, R., Kalla, S. L., Tang, Y., & Huang, J. (2011). The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications, 62(3), 902–917. https://doi.org/10.1016/J.CAMWA.2011.03.054
20. Singh, S., Han, S., & Lee, S. M. (2021). Adaptive single input sliding mode control for hybrid-synchronization of uncertain hyperchaotic Lu systems. Journal of the Franklin Institute. https://doi.org/10.1016/J.JFRANKLIN.2021.07.037
21. Vaidyanathan, S., Sambas, A., Mujiarto, Mamat, M., Wilarso, Mada Sanjaya, W. S., Sutoni, A., & Gunawan, I. (2021). A New 4-D Multistable Hyperchaotic Two-Scroll System, its Bifurcation Analysis, Synchronization and Circuit Simulation. Journal of Physics: Conference Series, 1764(1). https://doi.org/10.1088/1742-6596/1764/1/012206
22. Wang, F., Wang, R., Iu, H. H. C., Liu, C., & Fernando, T. (2019). A Novel Multi-Shape Chaotic Attractor and Its FPGA Implementation. IEEE Transactions on Circuits and Systems II: Express Briefs, 66(12), 2062–2066. https://doi.org/10.1109/TCSII.2019.2907709
23. Wang, J., Yu, W., Wang, J., Zhao, Y., Zhang, J., & Jiang, D. (2019). A new six-dimensional hyperchaotic system and its secure communication circuit implementation. International Journal of Circuit Theory and Applications, 47(5), 702–717. https://doi.org/10.1002/CTA.2617
24. Wang, P., Wen, G., Yu, X., Yu, W., & Huang, T. (2019). Synchronization of multi-layer networks: From node-to-node synchronization to complete synchronization. IEEE Transactions on Circuits and Systems I: Regular Papers, 66(3), 1141–1152. https://doi.org/10.1109/TCSI.2018.2877414
25. Wang, S., Hong, L., Jiang, J., & Li, X. (2020). Synchronization precision analysis of a fractional-order hyperchaos with application to image encryption. AIP Advances, 10(10). https://doi.org/10.1063/5.0012493
26. Wu, X., Lu, H., & Shen, S. (2009). Synchronization of a new fractional-order hyperchaotic system. Physics Letters, Section A: General, Atomic and Solid State Physics, 373(27–28), 2329–2337. https://doi.org/10.1016/j.physleta.2009.04.063
27. Yılmaz, G., Altun, K., & Günay, E. (2022). Synchronization of hyperchaotic Wang-Liu system with experimental implementation on FPAA and FPGA. Analog Integrated Circuits and Signal Processing, 113(2), 145–161. https://doi.org/10.1007/S10470-022-02073-4

There are 27 citations in total.

Details

Primary Language	English
Subjects	Electronics
Journal Section	Research Articles
Authors	Gülnur Yılmaz 0000-0001-8940-7883 Enis Günay 0000-0002-0447-6810
Early Pub Date	March 28, 2024
Publication Date	April 22, 2024
Submission Date	August 9, 2023
Acceptance Date	January 9, 2024
Published in Issue	Year 2024 Volume: 29 Issue: 1

Cite

APA	Yılmaz, G., & Günay, E. (2024). FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi, 29(1), 85-100. https://doi.org/10.17482/uumfd.1339620
AMA	Yılmaz G, Günay E. FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION. UUJFE. April 2024;29(1):85-100. doi:10.17482/uumfd.1339620
Chicago	Yılmaz, Gülnur, and Enis Günay. “FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 29, no. 1 (April 2024): 85-100. https://doi.org/10.17482/uumfd.1339620.
EndNote	Yılmaz G, Günay E (April 1, 2024) FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 29 1 85–100.
IEEE	G. Yılmaz and E. Günay, “FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION”, UUJFE, vol. 29, no. 1, pp. 85–100, 2024, doi: 10.17482/uumfd.1339620.
ISNAD	Yılmaz, Gülnur - Günay, Enis. “FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi 29/1 (April 2024), 85-100. https://doi.org/10.17482/uumfd.1339620.
JAMA	Yılmaz G, Günay E. FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION. UUJFE. 2024;29:85–100.
MLA	Yılmaz, Gülnur and Enis Günay. “FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION”. Uludağ Üniversitesi Mühendislik Fakültesi Dergisi, vol. 29, no. 1, 2024, pp. 85-100, doi:10.17482/uumfd.1339620.
Vancouver	Yılmaz G, Günay E. FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION. UUJFE. 2024;29(1):85-100.

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