# American Institue of Mathematical Sciences

2018, 38(2): 889-904. doi: 10.3934/dcds.2018038

## Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Facultat de Ciències, 08193 Bellaterra, Spain 2 Departament de Matemàtiques, Universitat Politècnica de Catalunya, Colom 1,08222 Terrassa, Spain

* Corresponding author: Víctor Mañosa

Received  January 2017 Revised  September 2017 Published  February 2018

Fund Project: The authors are supported by Ministry of Economy, Industry and Competitiveness of the Spanish Government through grants MINECO/FEDER MTM2016-77278-P (first and second authors) and DPI2016-77407-P AEI/FEDER, UE (third author). The first and second authors are also supported by the grant 2014-SGR-568 from AGAUR, Generalitat de Catalunya. The third author is supported by the grant 2014-SGR-859 from AGAUR, Generalitat de Catalunya

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.

Citation: Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
##### References:
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Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [13] F. M. Dannan, S. Elaydi, V. Ponomarenko, Stability of hyperbolic and nonhyperbolic fixed points of one-dimensional maps, J. Difference Equations and Appl., 9 (2003), 449-457. doi: 10.1080/1023619031000078315. [14] S. Elaydi, An Introduction to Difference Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1. [15] S. Elaydi, R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273. doi: 10.1016/j.jde.2003.10.024. [16] S. Elaydi, R. J. Sacker, Periodic difference equations, population biology and the Cushing-Henson conjectures, Math. Biosci., 201 (2006), 195-207. doi: 10.1016/j.mbs.2005.12.021. [17] J. E. Franke, J. F. Selgrade, Attractors for discrete periodic dynamical systems, J. Math. Anal. Appl., 286 (2003), 64-79. doi: 10.1016/S0022-247X(03)00417-7. [18] G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo's paradox, Nature, 402 (1999), p864. [19] W. P. Johnson, The curious history of Faá di Bruno's formula, Amer. Math. Monthly, 109 (2002), 217-234. doi: 10.2307/2695352. [20] R. Jungers, The Joint Spectral Radius. Theory and Applications, Lecture Notes in Control and Information Sciences 385, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-95980-9. [21] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (PA), 1976. [22] R. McGehee, A stable manifold theorem for degenerated fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6. [23] J. M. R. Parrondo, How to cheat a bad mathematician, Part of the presentation given in EEC HC&M Network on Complexity and Chaos (\#ERBCHRX-CT940546), ISI, Torino, Italy (1996), Unpublished. Available from: http://seneca.fis.ucm.es/parr/GAMES/cheat.pdf. Accessed September 4,2017. [24] R. Roy and F. W. Olver, Elementary functions: Lambert W-function, in NIST Handbook of Mathematical Functions (eds. F.W. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark), Cambridge University Press, Chapter 4, (2010), 103{134. Available from: http://dlmf.nist.gov/4.13. Accessed September 4,2017. [25] R. J. Sacker, H. von Bremen, A conjecture on the stability of periodic solutions of Ricker's equation with periodic parameters, Appl. Math. Comp., 217 (2010), 1213-1219. doi: 10.1016/j.amc.2010.05.049. [26] J. F. Selgrade, J. H. Roberds, On the structure of attractors for discrete, periodically forced systems with applications to population models, Physica D, 158 (2001), 69-82. doi: 10.1016/S0167-2789(01)00324-4. [27] J. F. Selgrade, J. H. Roberds, Global attractors for a discrete selection model with periodic immigration, J. Difference Equations and Appl., 13 (2007), 275-287. doi: 10.1080/10236190601079100. [28] C. Simó, Stability of parabolic points of area preserving analytic diffeomorphisms, Proceedings of the seventh Spanish-Portuguese conference on mathematics, Part Ⅲ (Sant Feliu de Guíxols, 1980) Publ. Sec. Mat. Univ. Autònoma Barcelona, 22 (1980), 67-70. [29] D. L. Slotnick, Asymptotic behavior of solutions of canonical systems near a closed, unstable orbit, in Contributions to the Theory of Nonlinear Oscillations (ed. S. Lefshetz), Annals of Mathematics Studies, no. 41 Princeton University Press, Princeton (NJ), (1958), 85-110. [30] F. Takens, Normal forms for certain singularities of vector fields, Annales Inst. Fourier, 23 (1973), 163-195. doi: 10.5802/aif.467. [31] J. Wright, Periodic systems of population models and enveloping functions, Comp. Math. Appl., 66 (2013), 2178-2195. doi: 10.1016/j.camwa.2013.08.013.

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##### References:
 [1] Z. AlSharawi, A global attractor in some discrete contest competition models with delay under the effect of periodic stocking Abstr. Appl. Anal. , 2013 (2013), Art. ID 101649, 7 pp. doi: 10.1155/2013/101649. [2] D. K. Arrowsmith and C. M. Place. An introduction to Dynamical Systems, Cambridge University Press, Cambridge, 1990. [3] I. Baldomá, E. Fontich, Stable manifolds associated to fixed points with linear part equal to the identity, J. Differential Equations, 197 (2004), 45-72. doi: 10.1016/j.jde.2003.07.005. [4] W.-J. Beyn, T. Hüls, M. C. Samtenschnieder, On $r$-periodic orbits of $k$-periodic maps, J. Difference Equations and Appl, 14 (2008), 865-887. doi: 10.1080/10236190801940010. [5] V. D. Blondel, J. Theys, J. N. Tsitsiklis, When is a pair of matrices stable?, in Unsolved Problems in Mathematical Systems and Control Theory (eds. V.D. Blondel, A. Megretski), Princeton Univ. Press, (2004), 304-308. [6] E. Camouzis, G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl., 333 (2007), 117-127. doi: 10.1016/j.jmaa.2006.10.096. [7] J. S. Cánovas, A. Linero, D. Peralta-Salas, Dynamic Parrondo's paradox, Physica D, 218 (2006), 177-184. doi: 10.1016/j.physd.2006.05.004. [8] K.-T. Chen, Normal forms of local diffeomorphisms on the real line, Duke Math. J., 35 (1968), 549-555. doi: 10.1215/S0012-7094-68-03556-4. [9] G. Chen, J. Della Dora, Normal forms for differentiable maps, Numerical Algorithms, 22 (1999), 213-230. doi: 10.1023/A:1019115025764. [10] A. Cima, A. Gasull and V. Mañosa, Global periodicity conditions for maps and recurrences via normal forms Int. J. Bifurcations and Chaos, 23 (2013), 1350182 (18 pages). doi: 10.1142/S0218127413501824. [11] A. Cima, A. Gasull, V. Mañosa, Non-integrability of measure preserving maps via Lie symmetries, J. Differential Equations, 259 (2015), 5115-5136. doi: 10.1016/j.jde.2015.06.019. [12] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. [13] F. M. Dannan, S. Elaydi, V. Ponomarenko, Stability of hyperbolic and nonhyperbolic fixed points of one-dimensional maps, J. Difference Equations and Appl., 9 (2003), 449-457. doi: 10.1080/1023619031000078315. [14] S. Elaydi, An Introduction to Difference Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1. [15] S. Elaydi, R. J. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differential Equations, 208 (2005), 258-273. doi: 10.1016/j.jde.2003.10.024. [16] S. Elaydi, R. J. Sacker, Periodic difference equations, population biology and the Cushing-Henson conjectures, Math. Biosci., 201 (2006), 195-207. doi: 10.1016/j.mbs.2005.12.021. [17] J. E. Franke, J. F. Selgrade, Attractors for discrete periodic dynamical systems, J. Math. Anal. Appl., 286 (2003), 64-79. doi: 10.1016/S0022-247X(03)00417-7. [18] G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo's paradox, Nature, 402 (1999), p864. [19] W. P. Johnson, The curious history of Faá di Bruno's formula, Amer. Math. Monthly, 109 (2002), 217-234. doi: 10.2307/2695352. [20] R. Jungers, The Joint Spectral Radius. Theory and Applications, Lecture Notes in Control and Information Sciences 385, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-95980-9. [21] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (PA), 1976. [22] R. McGehee, A stable manifold theorem for degenerated fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6. [23] J. M. R. Parrondo, How to cheat a bad mathematician, Part of the presentation given in EEC HC&M Network on Complexity and Chaos (\#ERBCHRX-CT940546), ISI, Torino, Italy (1996), Unpublished. Available from: http://seneca.fis.ucm.es/parr/GAMES/cheat.pdf. Accessed September 4,2017. [24] R. Roy and F. W. Olver, Elementary functions: Lambert W-function, in NIST Handbook of Mathematical Functions (eds. F.W. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark), Cambridge University Press, Chapter 4, (2010), 103{134. Available from: http://dlmf.nist.gov/4.13. Accessed September 4,2017. [25] R. J. Sacker, H. von Bremen, A conjecture on the stability of periodic solutions of Ricker's equation with periodic parameters, Appl. Math. Comp., 217 (2010), 1213-1219. doi: 10.1016/j.amc.2010.05.049. [26] J. F. Selgrade, J. H. Roberds, On the structure of attractors for discrete, periodically forced systems with applications to population models, Physica D, 158 (2001), 69-82. doi: 10.1016/S0167-2789(01)00324-4. [27] J. F. Selgrade, J. H. Roberds, Global attractors for a discrete selection model with periodic immigration, J. Difference Equations and Appl., 13 (2007), 275-287. doi: 10.1080/10236190601079100. [28] C. Simó, Stability of parabolic points of area preserving analytic diffeomorphisms, Proceedings of the seventh Spanish-Portuguese conference on mathematics, Part Ⅲ (Sant Feliu de Guíxols, 1980) Publ. Sec. Mat. Univ. Autònoma Barcelona, 22 (1980), 67-70. [29] D. L. Slotnick, Asymptotic behavior of solutions of canonical systems near a closed, unstable orbit, in Contributions to the Theory of Nonlinear Oscillations (ed. S. Lefshetz), Annals of Mathematics Studies, no. 41 Princeton University Press, Princeton (NJ), (1958), 85-110. [30] F. Takens, Normal forms for certain singularities of vector fields, Annales Inst. Fourier, 23 (1973), 163-195. doi: 10.5802/aif.467. [31] J. Wright, Periodic systems of population models and enveloping functions, Comp. Math. Appl., 66 (2013), 2178-2195. doi: 10.1016/j.camwa.2013.08.013.
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