Generalized Form of Parrondo's Paradoxical Game with Applications to Chaos Control

In this paper, we show that a generalized form of Parrondo's paradoxical game can be applied to discrete systems, working out the logistic map as a concrete example, to generate stable orbits. Written in Parrondo's terms, this reads: chaos1 + chaos2 + ⋯ + chaosN = order, where chaosi, i = 1, 2, …, N, are denoted as the chaotic behaviors generated by N values of the parameter control, and by order one understands some stable behavior. The numerical results are sustained by quantitative dynamics generated by Parrondo's game. The implementation of the generalized Parrondo's game is realized here via the parameter switching (PS) algorithm for continuous-time systems [Danca, 2013] adapted to the logistic map. Some related results for more general maps on averaging, which represent discrete analogies of the PS method for ODE, are also presented and discussed.

[1]  Marius-F. Danca,et al.  Finding attractors of continuous-time systems by parameter switching , 2011, 1102.2178.

[2]  Marius-F. Danca.  Convergence of a parameter switching algorithm for a class of nonlinear continuous systems and a generalization of Parrondo's paradox , 2013, Commun. Nonlinear Sci. Numer. Simul..

[3]  Rajeev Kohli,et al.  Why Parrondo's paradox is irrelevant for utility theory, stock buying, and the emergence of life , 2003, Complex..

[4]  Marius-F. Danca,et al.  Alternated Julia Sets and Connectivity Properties , 2009, Int. J. Bifurc. Chaos.

[5]  Abbott,et al.  New paradoxical games based on brownian ratchets , 2000, Physical review letters.

[6]  Derek Abbott,et al.  Asymmetry and disorder: A decade of Parrondo's Paradox , 2010 .

[7]  John L. Casti.  Losing games for your winning play , 2001, Complex..

[8]  David Heath,et al.  Discrete and continuous ratchets: from coin toss to molecular motor , 2002 .

[9]  Derek Abbott,et al.  Game theory: Losing strategies can win by Parrondo's paradox , 1999, Nature.

[10]  Derek Abbott,et al.  Minimal Brownian ratchet: an exactly solvable model. , 2002, Physical review letters.

[11]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[12]  Katja Gruenewald,et al.  Theory Of Games And Statistical Decisions , 2016 .

[13]  Marius-F. Danca,et al.  An averaging model for chaotic system with periodic time-varying parameter , 2010, Appl. Math. Comput..

[14]  R. Toral,et al.  Discrete–time ratchets, the Fokker–Planck equation and Parrondo's paradox , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  H. Scott Dumas,et al.  First-Order Averaging Principles for Maps with Applications to Accelerator Beam Dynamics , 2004, SIAM J. Appl. Dyn. Syst..

[16]  S. Elaydi.  An introduction to difference equations , 1995 .

[17]  J. Almeida,et al.  Can two chaotic systems give rise to order , 2004, nlin/0406010.

[18]  Marius-F. Danca,et al.  Parrondo's Game Model to Find numerically Stable attractors of a Tumor Growth Model , 2012, Int. J. Bifurc. Chaos.

[19]  Derek Abbott,et al.  Parrondo's paradox , 1999 .

[20]  Wang Fulai,et al.  Improvement and empirical research on chaos control by theory of "chaos + chaos = order". , 2012, Chaos.