Dynamic Analysis of The Fractional-Order Sprott-K Chaotic System and FPGA Implementation

Abstract

In this article, the fractional-order analysis and experimental application of the Sprott-K chaotic system using Field Programmable Gate Array (FPGA) hardware is presented. In the study, the attractor structures of the Sprott-K chaotic system, which were first obtained with the Simulink model, were realized. Mathematical analysis of Sprott-K dynamic equations was made and the minimum fractional-order at which the dynamic system entered chaos was determined. The integer-order chaotic behavior of the Sprott-K chaotic system is compared with the Simulink simulation of the minimum fractional-order system. Fractional-order analysis of the system was carried out using Carlson method, one of the rational approximation models. Transfer functions in the frequency domain are obtained for the fractional-order in which the system goes into chaos with the Carlson method. Fractional-order transfer functions in the frequency domain obtained have been converted to the discrete time z transfer function. The FPGA design of the system was designed using the dynamic structure Simulink and the code conversion was performed using MATLAB's HDL code compiler. Chaotic system was realized by downloading the bitstream file obtained from the compiler to the Xilinx FPGA ZedBoard Zynq-7000 chip. The results show that FPGA structures provide the desired accuracy and high speed realizations for fractional-order chaotic systems.

Keywords

• Chaos
• fractional-order
• Carlson method
• Sprott-K
• FPGA
• HDL coder.

Kesir Dereceli Sprott-K Kaotik Sisteminin Dinamik Analizi ve FPGA Uygulaması

Öz

Bu makalede, Alanda Programlanabilir Kapı Dizileri (Field Programmable Gate Array, FPGA) donanımı kullanılarak Sprott-K kaotik sisteminin kesir dereceli analiz ve deneysel uygulaması sunulmaktadır. Çalışmada, Sprott-K kaotik sisteminin ilk olarak Simulink modeli ile elde edilen çeker yapıları gerçekleştirilmiştir. Sprott-K dinamik denklemlerin matematiksel analizleri yapılarak dinamik sistemin kaosa girdiği minimum kesir derecesi belirlenmiştir. Sprott-K kaotik sisteminin tam dereceli kaotik davranışı minimum kesir dereceli sistem ile Simulink benzetimi karşılaştırılmıştır. Sistemin kesir dereceli analizi rasyonel yaklaşım modellerinden Carlson metodu kullanılarak gerçekleştirilmiştir. Carlson metodu ile sistemin kaosa girdiği kesir derecesi için frekans domenindeki transfer fonksiyonları elde edilmiştir. Elde edilen frekans domenindeki kesir dereceli transfer fonksiyonları ayrık zaman z transfer fonksiyonuna çevrilmiştir. Sistemin FPGA tasarımı, dinamik yapı Simulink kullanılarak tasarlanmış ve MATLAB'ın HDL kod derleyicisi kullanılarak kod dönüşümü gerçekleştirilmiştir. Kaotik sistem, derleyiciden elde edilen bit akışı dosyası Xilinx FPGA ZedBoard Zynq-7000 yongasına indirilerek gerçekleştirilmiştir. Sonuçlar, FPGA yapılarının kesir dereceli kaotik sistemler için istenen doğruluk ve yüksek hızlı gerçekleştirmeler sağladığını göstermektedir.

Anahtar Kelimeler

• Kaos
• kesir-derecesi
• Carlson metodu
• Sprott-K
• FPGA
• HDL kodlayıcı.

References

1. Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1997). Chaos in differential equations. In Chaos (pp. 359-397). Springer, Berlin, Heidelberg.
2. Alvarez, G., & Li, S. (2006). Some basic cryptographic requirements for chaos-based cryptosystems. International journal of bifurcation and chaos, 16(08), 2129-2151.
3. Caponetto, R., Dongola, G., Maione, G., & Pisano, A. (2014). Integrated technology fractional order proportional-integral-derivative design. Journal of Vibration and Control, 20(7), 1066-1075.
4. Carlson, G., & Halijak, C. (1964). Approximation of fractional capacitors (1/s)^(1/n) by a regular Newton process. IEEE Transactions on Circuit Theory, 11(2), 210-213.
5. Charef, A., Sun, H. H., Tsao, Y. Y., & Onaral, B. (1992). Fractal system as represented by singularity function. IEEE Transactions on automatic Control, 37(9), 1465-1470.
6. Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and chaos, 9(07), 1465-1466.
7. Chua, L. O. (1992). The genesis of Chua's circuit. Berkeley, CA, USA: Electronics Research Laboratory, College of Engineering, University of California.
8. Günay, E., & Altun, K. (2018). Güvenilir Haberleşmede Açık Kapalı Kaotik Anahtarlama Sisteminin FPGA Kullanılarak Gerçekleştirilmesi. Selçuk Üniversitesi Mühendislik, Bilim ve Teknoloji Dergisi, 6(4), 559-571.

9. Holmes, P. (1979). A nonlinear oscillator with a strange attractor. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 292(1394), 419-448.
10. Holmes, P. (1990). Poincaré, celestial mechanics, dynamical-systems theory and “chaos”. Physics Reports, 193(3), 137-163.
11. Howard, R. M. (2004). Principles of random signal analysis and low noise design: The power spectral density and its applications. John Wiley & Sons.
12. Huang, X., Zhang, B., Qin, H., & An, W. (2017). Closed-form design of variable fractional-delay FIR filters with low or middle cutoff frequencies. IEEE Transactions on Circuits and Systems I: Regular Papers, 65(2), 628-637.
13. Jiang, C. X., Carletta, J. E., & Hartley, T. T. (2007). Implementation of fractional-order operators on field programmable gate arrays. In Advances in fractional calculus (pp. 333-346). Springer, Dordrecht.
14. Jiang, C. X., Carletta, J. E., Hartley, T. T., & Veillette, R. J. (2013). A systematic approach for implementing fractional-order operators and systems. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), 301-312.
15. Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of atmospheric sciences, 20(2), 130-141.
16. Matignon, D. (1996, July). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications (Vol. 2, No. 1, pp. 963-968).
17. Miller, K. S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.
18. Oldham, K., & Spanier, J. (1974). The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
19. Oliveira Valério, D. P. M. (2005). Fractional robust system control. Universidade Técnica de Lisboa.
20. Peitgen, H. O., Jürgens, H., & Saupe, D. (2006). Chaos and fractals: new frontiers of science. Springer Science & Business Media.
21. Petráš, I. (2011). Fractional-order chaotic systems. In Fractional-order nonlinear systems (pp. 103-184). Springer, Berlin, Heidelberg.
22. Petráš, I. (2011). Fractional-order nonlinear systems: modeling, analysis and simulation. Springer Science & Business Media.
23. Podlubny, I. (1999). An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering, 198, xxiv+-340.
24. Rajagopal, K., Akgul, A., Jafari, S., & Aricioglu, B. (2018). A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications. Nonlinear Dynamics, 91(2), 957-974.
25. Ross, B. (1977). The development of fractional calculus 1695–1900. Historia Mathematica, 4(1), 75-89.
26. Rössler, O. E. (1976). An equation for continuous chaos. Physics Letters A, 57(5), 397-398.
27. Shah, D. K., Chaurasiya, R. B., Vyawahare, V. A., Pichhode, K., & Patil, M. D. (2017). FPGA implementation of fractional-order chaotic systems. AEU-International Journal of Electronics and Communications, 78, 245-257.
28. Sprott, J. C. (1994). Some simple chaotic flows. Physical review E, 50(2), R647.
29. Tolba, M. F., AbdelAty, A. M., Soliman, N. S., Said, L. A., Madian, A. H., Azar, A. T., & Radwan, A. G. (2017). FPGA implementation of two fractional order chaotic systems. AEU-International Journal of Electronics and Communications, 78, 162-172.
30. Vaidyanathan, S. (2016). Generalized projective synchronization of vaidyanathan chaotic system via active and adaptive control. In Advances and Applications in Nonlinear Control Systems (pp. 97-116). Springer, Cham.
31. Zhang, Y., Liu, Z., & Zheng, X. (2008, December). A chaos-based image encryption ASIC using reconfigurable logic. In APCCAS 2008-2008 IEEE Asia Pacific Conference on Circuits and Systems (pp. 1782-1785). IEEE.