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Complex Systems and Networks pp 263–282Cite as

Chaos Control and Anticontrol of Complex Systems via Parrondo’s Game

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Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter, we prove analytically and numerically aided by computer simulations, that the Parrondo game can be implemented numerically to control and anticontrol chaos of a large class of nonlinear continuous-time and discrete-time systems. The game states that alternating loosing gains of two games, one can actually obtain a winning game, i.e.: “losing \(+\) losing \(=\) winning” or, in other words: “two ugly parents can have beautiful children” (Zeilberger, on receiving the 1998 Leroy P. Steele Prize). For this purpose, the Parameter Switching (PS) algorithm is implemented. The PS algorithm switches the control parameter of the underlying system, within a set of values as the system evolves. The obtained attractor matches the attractor obtained by replacing the parameter with the average of switched values. The systems to which the PS algorithm based Parrondo’s game applies are continuous-time of integer or fractional order ones such as: Lorenz system, Chen system, Chua system, Rössler system, to name just a few, and also discrete-time systems and fractals. Compared with some other works on switch systems, the PS algorithm utilized in this chapter is a convergent algorithm which allows to approximate any desired dynamic to arbitrary accuracy.

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Notes

  1. 1.

    Also, a and c can be considered as control parameters to match to the form (10.7).

  2. 2.

    There exists no convergence result so far. However, intensively numerical tests reveal, like in the considered example, a good match between the switched attractor and the averaged attractor in the case of fractional-order systems.

  3. 3.

    E.g. the pseudorandom function, found in all dedicated software.

  4. 4.

    In [3, 23], some particular forms of switches are used to study the behavior of alternated orbits for the more accessible quadratic (Mandelbrot) map \(x_{k+1}=x_k^2+p\).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emanuel University of Oradea, 410597, Oradea, Romania

    Marius-F. Danca

  2. Romanian Institute of Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

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  1. Marius-F. Danca
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Correspondence to Marius-F. Danca .

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Editors and Affiliations

  1. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Jinhu Lü

  2. School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia

    Xinghuo Yu

  3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China

    Guanrong Chen

  4. Department of Mathematics, Southeast University, Nanjing, China

    Wenwu Yu

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Danca, MF. (2016). Chaos Control and Anticontrol of Complex Systems via Parrondo’s Game. In: Lü, J., Yu, X., Chen, G., Yu, W. (eds) Complex Systems and Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47824-0_10

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  • DOI: https://doi.org/10.1007/978-3-662-47824-0_10

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