Abstract
Scholars have done extensive research on dynamic analysis and analog circuits implementation of the classical fractional order chaotic system-Liu System (FOLS). However, they did not rigorously prove the existence of FOLS from the perspective of mathematics. And they also did not effectively design digital circuits to generate signals of the fractional order chaotic systems, especially the 2.7-order system. This paper selects an appropriate Poincare section where a first return Poincare map of FOLS was defined. Based on computer-assisted verification method, the conclusion is that the Poincare map is semi-conjugate to a 2-shift map and the topological entropy of the map is no less than ln 2, which rigorously verifies the existence of chaotic behavior in the 2.7-order Liu system. This proof is necessary before the chaotic system is used for information encryption. The next and most significant task is to build a system model through DSP-Builder software and generate chaotic signals using Field Programmable Gate Array chip. The results of oscilloscope consistent with numerical simulations, which lays the foundation for image and video streaming encryption.
Highlights
In recent years, fractional calculus has been a hot topic in various academic disciplines
Research on integer-order chaotic systems [11] has been extended to fractional-order chaotic systems, such as the fractional Chen system [12], fractional Lorenz system [13], [14], The associate editor coordinating the review of this manuscript and approving it for publication was Remigiusz Wisniewski
Dong et al.: Topological Horseshoe Analysis and FPGA Implementation of a Classical Fractional Order Chaotic System analytical method which can verify the existence of chaos from a mathematical point of view [19], [20]
Summary
Fractional calculus has been a hot topic in various academic disciplines. E. Dong et al.: Topological Horseshoe Analysis and FPGA Implementation of a Classical Fractional Order Chaotic System analytical method which can verify the existence of chaos from a mathematical point of view [19], [20]. The topological horseshoe lemma was proposed as a practical and useful computer-assisted proof method [21] This theory has already been used to prove the existence of chaos in some integer-order systems, including typical 3D systems [22]–[25] and hyperchaos systems [26]–[29]. The contribution of this paper is that based on the method of topological horseshoe analysis adopted in many integer-order chaotic systems, it is proved that a topological horseshoe does exist in a 2.7-order Liu system, which verifies its chaotic characteristic.
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