Digitally generating true orbits of binary shift chaotic maps and their conjugates

Abstract

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-off and 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-off and 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.