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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Also, a and c can be considered as control parameters to match to the form (10.7).
- 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.
E.g. the pseudorandom function, found in all dedicated software.
- 4.
References
Abbot, D.: Assymmetry and disorder: a decade of parrondo’s paradox. Fluct. Noise Lett. 9, 129–156 (2010)
Abbott, D., Davies, P.C.W., Shalizi, C.R.: Order from disorder: the role of noise in creative processes: a special issue on game theory and evolutionary processes -overview. Fluct. Noise Lett. 7(2), 1–12 (2003)
Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order. Phys. D. 200, 124–132 (2005)
Amengual, P., Allison, A., Toral, R., Abbott, D.: Discrete-time ratchets, the Fokker-Planck equation and parrondo’s paradox. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460, 2269–2284 (2004)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Singapore (2012)
Blackwell, D., Girshick, M.A.: Theory of Games and Statistical Decisions. John Wiley, New York (1954)
Blanchard, P., Devaney, R., Keen, L.: Complex dynamics and symbolic dynamics, symbolic dynamics and Its Applications. In: Williams, S.G. (ed.) Proceedings of Symposium in Applied Mathematics 60, pp. 37–60 (2004)
Blanchard, P.: Disconnected Julia sets. In: Barnsley, M., Demko, S. (eds.) Chaotic Dynamics and Fractals, pp. 181–201. Academic Press, San Diego (1986)
Branner, B., Hubbard, J.H.: Iteration of cubic polynomials, part II: patterns and parapatterns. Acta Math. 169, 229–325 (1992)
Buceta, J., Lindenberg, K., Parrondo, J.M.R.: Spatial patterns induced by random switching. Fluct. Noise Lett. 2(1), L21–L29 (2002)
Buceta, J., Lindenberg, K., Parrondo, J.M.R.: Stationary and oscillatory spatial patterns induced by global periodic switching. Phys. Rev. Lett. 88, 024103 (2002)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent–II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967) (reprinted in Fract. Calc. Appl. Anal.) 10(3), 309–324 (2007)
Li, C., Chen, G.: Chaos in the fractional order Chen system and its control. Chaos Soliton. Fract. 22(3), 549–554 (2004)
Danca, M.F., Fečkan, M., Romera, M.: Generalized Form of Parrondo’s Paradoxical Game with Applications to Chaos Control. Int. J. Bifurcat. Chaos. 2014, Accepted
Danca, M.F., Bourke, P., Romera, M.: Graphical exploration of the connectivity sets of alternated Julia sets; M, the set of disconnected alternated Julia sets. Nonlinear Dynam. 73, 1155–1163 (2013)
Danca, M.F., Wallace, K.S.: Tang, Chen, G.: A switching scheme for synthesizing attractors of dissipative chaotic systems. Appl. Math. Comput. 201(1–2), 650–667 (2008)
Danca, M.F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurcat. Chaos. 19(6), 2123–2129 (2009)
Davies, P.C.W.: Physics and life: The Abdus Salam Memorial Lecture. 6th Trieste Conference on Chemical Evolution. Eds: J. Chela-Flores, T. Tobias, and F. Raulin. Kluwer Academic Publishers, 13–20 (2001)
Drebin, R.A., Carpenter, L., Hanrahan, P.: Volume rendering. In: Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 88), Vol. 22 pp. 65–74 (1988)
Fatou, P.: Sur les équations fonctionnelles. B. Soc. Math. Fr. 47 (1919) 161–271; 48 (1920) 33–94; 48 (1920) 208–314. http://smf.emath.fr/Publications/Bulletin/
Fečan, M., Danca, M.F.: Note on a Parameter Switching Method for Nonlinear ODEs. Math. Slovaca. Accepted (2014)
Foias, C., Jolly, M.S.: On the numerical algebraic approximation of global attractors. Nonlinearity 8, 295–319 (1995)
Fulai, W.: Improvement and empirical research on chaos control by theory of “chaos+chaos=order”. CHAOS 22, 043145 (2012)
Groeber, P.: On Parrondos games as generalized by behrends. Lect. Notes Cont. Inform. Sci. 341, 223–230 (2006)
Hale, J.K.: Ordinary Differential Equations. Dover Publications, New York (2009) (first published: John Wiley & Sons (1969)
Heath, D., Kinderlehrer, D., Kowalczyk, M.: Discrete and continuous ratchets: from coin toss to molecular motor. Discrete Cont. Dyn.-B. 2,153–167 (2002)
Julia, G.: Mémoire sur l’itération des fonctions rationnelles. J. Math. Pure Appl. 8, 47–245 (1918)
Lee, Y., Allison, A., Abbott, D., Stanley, H.E.: Minimal Brownian ratchet: an exactly solvable model. Phys. Rev. Lett. 91, 220601 (2003)
Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Soliton. Fract. 22(2), 443–450 (2004)
Li, C., Zeng, F.: Finite difference methods for fractional differential equations. Int. J. Bifurcat. Chaos. 22(4), 1230014 (2012)
Limaye, A.: Drishti: a volume exploration and presentation tool. In: Proceedings SPIE 8506, Developments in X-Ray Tomography VIII, 85060X (2012)
Mandelbrot, B.B.: The Fractal Geometry of Nature. Freeman, New York (1977)
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, New York (1974)
Parrondo, J.M.R., Harmer, G.P., Abbott, D.: New paradoxical games based on Brownian ratchets. Phys. Rev. Lett. 85, 5226–5229 (2000)
Peitgen, H.O., Saupe, D.: The Science of Fractal Images. Springer, New York (1988)
Percus, O.E., Percus, J.K.: Can two wrongs make a right? Coin-tossing games and parrondo’s paradox. Math. Intell. 24, 68–72 (2002)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Qiu, W., Yin, Y.: Proof of the Branner-Hubbard conjecture on Cantor Julia sets. Sci. China Ser. A. 52(1), 45–65 (2009)
Romera, M., Pastor, G., Danca, M.F., Martin, A., Orue, A.B., Montoya, F.: Alternate iteration of a quadratic map. Int. J. Bifurcat. Chaos. Accepted (2014)
Romera, M., Small, M., Danca, M.F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192(1), 283–297 (2007)
Scherer, R., Kalla, S.L., Tang, Y., Huang, J.: The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62(3), 902–917 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-47824-0_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-47823-3
Online ISBN: 978-3-662-47824-0
eBook Packages: EngineeringEngineering (R0)