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Random Numbers from a Delay Equation

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Abstract

Delay differential equations can have “chaotic” solutions that can be used to mimic Brownian motion. Since a Brownian motion is random in its velocity, it is reasonable to think that a random number generator might be constructed from such a model. In this preliminary study, we consider one specific example of this and show that it satisfies criteria commonly employed in the testing of random number generators (from TestU01’s very stringent “Big Crush” battery of tests). A technique termed digit discarding, commonly used in both this generator and physical RNGs using laser feedback systems, is discussed with regard to the maximal Lyapunov exponent. Also, we benchmark the generator to a contemporary common method: the multiple recursive generator, MRG32k3a. Although our method is about 7 times slower than MRG32k3a, there is in principle no apparent limit on the number of possible values that can be generated from the scheme we present here.

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Notes

  1. A solution of Eq. 3.3 is associated with a time sequence \(t_0< t_1< \cdots< t_n< \cdots \), which is defined such that \(\sin (2\pi \beta v(t)) \ge 0\) when \(t\in [t_{2k}, t_{2k+1})\), and \(\sin (2\pi \beta v(t)) < 0 \) when \(t\in [t_{2k-1}, t_{2k})\). Furthermore, if the sequence \((t_0, \cdots , t_n)\) is known, then the solution \(v(t)\) when \(t\in (t_n, t_n + 1)\) can be obtained explicitly, and therefore, \(t_{n+1}\), which is defined as \(\sin (2\beta v(t_{n+1})) = 0\), is determined by \((t_0, \cdots , t_n)\). Once we obtain the entire sequence \(\{t_n\}\), the solution of Eq. 3.3 consists of exponentially increasing or decreasing segments on each interval \([t_n, t_{n+1}]\). Nevertheless, the nature and properties of the map \(t_{n+1} = F_n(t_0, t_1,\cdots , t_{n})\) is still not characterized and has defied analysis to date.

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Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada). We would like to thank Dimitri Breda for providing the scripts that were used to calculate the Lyapunov exponents and Tony Humphries, Erik Van Vleck, Joshua Lackman, and Serhiy Yanchuk for their help.

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Authors and Affiliations

  1. Departments of Physiology, Physics, and Mathematics, and Centre for Applied Mathematics in Bioscience and Medicine (CAMBAM), McGill University, 3655 Promenade Sir William Osler, Montréal, QC, H3G 1Y6, Canada

    Julian Self & Michael C. Mackey

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  1. Julian Self
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  2. Michael C. Mackey
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Correspondence to Julian Self.

Additional information

Communicated by Govind Menon.

Deterministic differential delay equations are well known to sometimes have chaotic solutions that are unpredictable in spite of the fact that they approach either ensemble or trajectory limiting densities that are independent of initial conditions (functions). We show that this characteristic may be used effectively for producing a random number generator.

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Self, J., Mackey, M.C. Random Numbers from a Delay Equation. J Nonlinear Sci 26, 1311–1327 (2016). https://doi.org/10.1007/s00332-016-9306-9

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  • DOI: https://doi.org/10.1007/s00332-016-9306-9

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