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.

Kaynakça

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