Digitally generating true orbits of binary shift chaotic maps and their conjugates
Digitally generating true orbits of binary shift chaotic maps and their conjugates
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DOI：
10.1016/j.cnsns.2018.02.039
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发表时间：
2018-09-01
影响因子：
3.9
通讯作者：
Kilic, Recai
中科院分区：
文献类型：
--
作者：
Ozturk, Ismail;Kilic, Recai
在数字平台上使用传统的有限精度计算不可能获得混沌行为。所有这些认识最终都是周期性的。此外，由于舍入和截断误差，周期轨道的数字计算常常是错误的。由于这些误差，周期性轨道很快偏离真实轨道，最终成为几乎所有初始条件下都会发生的少数周期之一。因此，混沌系统的数字计算并不代表数学定义的原始系统的真实轨道。这种差异在伯努利图或帐篷图等二元平移混沌图的模拟中变得很明显。尽管这些系统是完美定义的混沌系统，但它们的数字实现总是收敛于零。在文献中，有一些研究用随机位代替最低有效零位来克服这个问题。在本文中，我们提出了使用这种简单方法来数字实现二进制移位混沌映射的算法。这些算法适用于软件和硬件解决方案，并且也适用于任何随机数生成器或重复位序列。根据随机数发生器的类型，可以得到映射的真周期轨道或真混沌轨道。此外，还表明，利用拓扑共轭，获得的二元移位混沌映射的真实轨道可用于计算其他映射的真实轨道，例如逻辑映射和切比雪夫映射，这些映射通常会受到舍入和截断误差的影响。使用所提出的算法在现场可编程门阵列（FPGA）平台上实现了二进制移位混沌图、逻辑图和切比雪夫图的硬件实现。 (c) 2018 Elsevier B.V. 保留所有权利。
It is impossible to obtain chaotic behavior using conventional finite precision calculations on a digital platform. All such realizations are eventually periodic. Also, digital calculations of the periodic orbits are often erroneous due to round-offand truncation errors. Because of these errors, periodic orbits quickly diverge from the true orbit and they end up into one of the few cycles that occur for almost all initial conditions. Hence, digital calculations of chaotic systems do not represent the true orbits of the mathematically defined original system. This discrepancy becomes evident in the simulations of the binary shift chaotic maps like Bernoulli map or tent map. Although these systems are perfectly well defined chaotic systems, their digital realizations always converge to zero. In the literature, there are some studies which replace the least significant zero bits by random bits to overcome this problem.In this paper, we propose the algorithms using this simple method for digitally implementing binary shift chaotic maps. These algorithms are suitable for both software and hardware solutions, and they are also applicable with any random number generator or a repeated bit sequence. According to the type of the random number generator, either true periodic orbits or true chaotic orbits of the map are obtained. Moreover, it is shown that, utilizing topological conjugacies, obtained true orbits of binary shift chaotic maps can be used to calculate true orbits of other maps such as logistic and Chebyshev maps which are normally subject to round-offand truncation errors. The hardware implementations of binary shift chaotic maps, logistic map and Chebyshev maps have been realized on a Field Programmable Gate Array (FPGA) platform using the proposed algorithms. (c) 2018 Elsevier B.V. All rights reserved.