#### For citation:

Anikin V. M., Inkin M. G., Plekhanov O. S. Preserving measure chaotic maps of domains in the form of rotation figures. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 1, pp. 90-103. DOI: 10.18500/0869-6632-2018-26-1-90-103

# Preserving measure chaotic maps of domains in the form of rotation figures

The aim of the article is to demonstrate an algorithm for constructing measure-preserving three-dimensional chaotic maps defined in domains formed by rotation bodies. On the one hand, the class of multidimensional chaotic mappings is expended, and on the other hand, we obtain formulas for simulating pseudorandom quantities that are in demand in problems solving by the Monte Carlo method. The analytical algorithm for constructing multidimensional maps consists of the following steps: 1) the presentation of the invariant density as the product of the unconditional distribution of a point coordinates of the map’s orbit and the conditional densities of the distribution of other coordinates (provided that the values of some coordinates take a fixed value); 2) finding the corresponding integral distribution laws for the coordinates of the point of the mapping; 3) presentation of the coordinates of the point of the orbit through pseudo-random variables by using the inverse function modeling method; 4) reduction of the obtained dependences to the form of chaotic mappings for a particular choice of a chaotic onedimensional map possessing a uniform invariant distribution. The last step allows us to present pseudorandom values as iterative deterministic procedures defined on areas of complex shape. Statistical properties correlate with an array of generated numbers that have the sense of the coordinates of a pseudorandom point in a space bounded by a rotation figure. Examples of the synthesis of three-dimensional chaotic mappings (generators of pseudorandom points) are considered both for the general case (defining the generator of the body of revolution by an arbitrary continuous function) and for specific types of threedimensional regions in the form of a sphere and a cone. Methods are discussed that allow one to smooth out the property of rationality set of computer numbers by modeling pseudo-random variables.

- Mikhailov G.A., Voitishok A V. Numerical Statistical Modeling. Monte Carlo Methods: Textbook. Allowance for students of universities. Moscow: Publishing Center «Academy», 2006, 368 p. (in Russian).
- Robert C., Casella G. Monte Carlo Statistical Methods. New York: Springer, 2004. 683 p.
- Sharakshane A.S., Zheleznov I.G., Ivnitsky V.A. Complex Systems. M.: High School, 1977. 247 p. (in Russian).
- Sobol I.M. Points Uniformly Filling a Multidimensional Cube. M.: Knowledge, 1985. 32 p. (in Russian).
- Smale S. Mathematical problems for the next century. The Mathematical Intelligencer, 1998, vol. 20, iss. 2, pp. 7–15.
- Kopytov N.P., Mityushov E.A. Mathematical model of reinforcing shells from fibrous composite materials and the problem of uniform distribution of points on surfaces. Bulletin of PSTU. Mechanics, 2010, iss 4, pp. 55–66 (in Russian).
- Kendall M.G., Moran P.A.P. Geometrical probability. London: Ch. Griffin & Company Ltd, 1963, 125 p.
- Kopytov N.P. The Monte Carlo method for estimating the expected neutralized surface area of a spherical viral particle randomly attacked by antibodies. Russian Journal of Biomechanics, 2012, vol. 16, iss. 3 (57), pp. 65–74 (in Russian).
- Billingsley P. Ergodic Theory and Information. New York: John Wiley and Sons, Inc., 1965, XIII+193 p.
- Anikin V.M., Goloubentsev A.F. Analytical Models of Deterministic Chaos. M.: Fizmatlit, 2007, 328 p. (in Russian).
- Ermakov S.M., Mikhailov G.A. Statistical Modeling. 2 ed., ext. M.: Fizmatlit, 1982, 296 p. (in Russian).
- Anikin V.M., Noyanova S.A. Two-dimensional chaotic mappings. Radio Engineering, 2005, iss. 4, pp. 63–70 (in Russian).
- Anikin V.M., Arkadaksky S.S., Remizov A.S. Non-Selfadjoined Linear Operators in Chaotic Dynamics. Saratov: Saratov University Publishing House, 2015. 96 p.
- Anikin V.M., Arkadaksky S.S., Kuptsov S.N., Remizov A.S. Polynomial eigenfunctions of the Perron–Frobenius operator. Izvestiya VUZ, Applied Nonlinear Dynamics, 2016, vol. 24, iss. 4, pp. 6–16 (in Russian).
- Anikin V.M. Spectral problems for the Perron–Frobenius operator. Izvestiya VUZ, Applied Nonlinear Dynamics, 2009, vol. 17, iss. 4, pp. 35–48 (in Russian).
- Anikin V.M., Remizov A.S., Arkadaksky S.S. The eigenfunctions and eigenvalues of the Perron–Frobenius operator of piecewise-linear chaotic mappings. Izvestiya VUZ, Applied Nonlinear Dynamics, 2007, vol. 15, iss. 2, pp. 62–75 (in Russian).
- Golubentsev A.F., Anikin V.M. Invariant functional subspaces of linear evolution operators of chaotic mappings. Izvestiya VUZ, Applied Nonlinear Dynamics, 2005, vol. 13, iss. 1–2, pp. 3–17 (in Russian).
- Baptista M.S. Cryptography with chaos. Phys. Lett., 1998, vol. A240, pp. 50–54.
- Anikin V.M., Chebanenko S.V. Chaotic mappings and coding of information: Modifications of the historically first algorithm. Heteromagnetic microelectronics, 2011, iss. 9, pp. 81–95 (in Russian).

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