Researchers effectively tune parameters of perturbation method to preserve chaos in output of Bernoulli displacement map – ScienceDaily

The Bernoulli displacement map is a well-known chaotic map in chaos theory. For a binary system, however, the output is not chaotic and instead reduces to zero. One way to prevent this is by breaking the state space of the map. In a new study, researchers explore one such perturbation method to obtain non-converging long-period outputs and analyze these periods using modular arithmetic, obtaining a complete list of parameter values ​​for optimal perturbations.

Is it possible for a deterministic system to be unpredictable? Although counterintuitive, the answer is yes. Such systems are called “chaotic systems”, which are characterized by sensitive dependence on initial conditions and long-term unpredictability. The behavior of such systems is often described using what is known as a “chaotic map”. Chaotic maps find applications in areas such as algorithm design, data analysis, and numerical simulations.

A well-known example of a chaotic map is the Bernoulli displacement map. In practical applications of the Bernoulli displacement map, the outputs are often required to have long periods. Oddly enough, however, when the Bernoulli shift map is implemented in a binary system such as a digital computer, the output sequence is no longer chaotic and instead converges to zero!

To this end, perturbation methods are an effective strategy where a perturbation is applied to the state of the Bernoulli displacement map to prevent its output from converging. However, the choice of parameters to obtain suitable perturbations has no theoretical basis.

In a recent study published online on October 21, 2022 and published in the journal Volume 165, Part 1 Chaos, solitons and fractals on December 2022, Professor Toru Ikeguchi of Tokyo University of Science in collaboration with Dr. Noriyoshi Sukegawa of the University of Tsukuba, both in Japan, is already addressing this problem by laying the theoretical foundations for efficient parameter tuning. “Although numerical simulations can tell us which parameter values ​​can prevent convergence, there is no theoretical basis for choosing these values. In this paper, we aimed to explore the theoretical support behind this choice,” explains Prof. Ikeguchi.

Accordingly, the researchers used modular arithmetic to set a dominant parameter in the perturbation method. In particular, they defined the best value for the parameter, which depended on the bit length specified in the implementations. The team further analyzed the baseline period for which the parameter had the best value. Their findings show that the resulting periods approach trivial theoretical upper bounds. Based on this, the researchers obtained a complete list of the best parameter values ​​for successful implementation of the Bernoulli displacement map.

Furthermore, an interesting consequence of their study is its relation to Artin’s conjecture on primitive roots, an open question in number theory. The researchers suggest that, provided Artin’s hypothesis is correct, their approach would be theoretically guaranteed to be effective for any bit length.

Overall, the theoretical foundations laid out in this study are of paramount importance for the practical applications of chaotic maps in general. “A notable advantage of our approach is that it provides theoretical support for choosing the best parameters. In addition, our analysis can be partially applied to other chaotic maps, such as the tent map and the logistic map,” emphasizes Dr. Sukegawa.

With various advantages such as simplicity and ease of implementation, Bernoulli displacement maps are highly desired in several practical applications. And as this study shows, sometimes chaos is preferable to order!

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Materials provided by Tokyo University of Science. Note: Content may be edited for style and length.

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