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

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

Anna Cima 1, , Armengol Gasull 1, and  Víctor Mañosa 2,,

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.

Keywords: Periodic discrete dynamical systems, non-hyperbolic points, local and global asymptotic stability, Parrondo's dynamic paradox.
Mathematics Subject Classification: Primary: 37C05, 37C25, 37C75; Secondary: 39A30.
Citation: Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
References:
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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.  Google Scholar

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[3]

I. Baldomá and 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.  Google Scholar

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W.-J. Beyn, T. Hüls and M. C. Samtenschnieder, On $r$-periodic orbits of $k$-periodic maps, J. Difference Equations and Appl, 14 (2008), 865-887.  doi: 10.1080/10236190801940010.  Google Scholar

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V. D. Blondel, J. Theys and 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.   Google Scholar

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E. Camouzis and G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl., 333 (2007), 117-127.  doi: 10.1016/j.jmaa.2006.10.096.  Google Scholar

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J. S. Cánovas, A. Linero and D. Peralta-Salas, Dynamic Parrondo's paradox, Physica D, 218 (2006), 177-184.  doi: 10.1016/j.physd.2006.05.004.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A. Cima, A. Gasull and 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.  Google Scholar

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R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750.  Google Scholar

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F. M. Dannan, S. Elaydi and 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.  Google Scholar

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S. Elaydi, An Introduction to Difference Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.  Google Scholar

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S. Elaydi and 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.  Google Scholar

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S. Elaydi and 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.  Google Scholar

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J. E. Franke and 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.  Google Scholar

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G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo's paradox, Nature, 402 (1999), p864. Google Scholar

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W. P. Johnson, The curious history of Faá di Bruno's formula, Amer. Math. Monthly, 109 (2002), 217-234.  doi: 10.2307/2695352.  Google Scholar

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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.  Google Scholar

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J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (PA), 1976.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

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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. Google Scholar

[25]

R. J. Sacker and 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.  Google Scholar

[26]

J. F. Selgrade and 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.  Google Scholar

[27]

J. F. Selgrade and 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

F. Takens, Normal forms for certain singularities of vector fields, Annales Inst. Fourier, 23 (1973), 163-195.  doi: 10.5802/aif.467.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

D. K. Arrowsmith and C. M. Place. An introduction to Dynamical Systems, Cambridge University Press, Cambridge, 1990.  Google Scholar

[3]

I. Baldomá and 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.  Google Scholar

[4]

W.-J. Beyn, T. Hüls and M. C. Samtenschnieder, On $r$-periodic orbits of $k$-periodic maps, J. Difference Equations and Appl, 14 (2008), 865-887.  doi: 10.1080/10236190801940010.  Google Scholar

[5]

V. D. Blondel, J. Theys and 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.   Google Scholar

[6]

E. Camouzis and G. Ladas, Periodically forced Pielou's equation, J. Math. Anal. Appl., 333 (2007), 117-127.  doi: 10.1016/j.jmaa.2006.10.096.  Google Scholar

[7]

J. S. Cánovas, A. Linero and D. Peralta-Salas, Dynamic Parrondo's paradox, Physica D, 218 (2006), 177-184.  doi: 10.1016/j.physd.2006.05.004.  Google Scholar

[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.  Google Scholar

[9]

G. Chen and J. Della Dora, Normal forms for differentiable maps, Numerical Algorithms, 22 (1999), 213-230.  doi: 10.1023/A:1019115025764.  Google Scholar

[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.  Google Scholar

[11]

A. Cima, A. Gasull and 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.  Google Scholar

[12]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750.  Google Scholar

[13]

F. M. Dannan, S. Elaydi and 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.  Google Scholar

[14]

S. Elaydi, An Introduction to Difference Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1.  Google Scholar

[15]

S. Elaydi and 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.  Google Scholar

[16]

S. Elaydi and 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.  Google Scholar

[17]

J. E. Franke and 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.  Google Scholar

[18]

G. P. Harmer and D. Abbott, Losing strategies can win by Parrondo's paradox, Nature, 402 (1999), p864. Google Scholar

[19]

W. P. Johnson, The curious history of Faá di Bruno's formula, Amer. Math. Monthly, 109 (2002), 217-234.  doi: 10.2307/2695352.  Google Scholar

[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.  Google Scholar

[21]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (PA), 1976.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

R. J. Sacker and 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.  Google Scholar

[26]

J. F. Selgrade and 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.  Google Scholar

[27]

J. F. Selgrade and 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

F. Takens, Normal forms for certain singularities of vector fields, Annales Inst. Fourier, 23 (1973), 163-195.  doi: 10.5802/aif.467.  Google Scholar

[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.  Google Scholar

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