American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
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 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.

[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.

[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. 

[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.

[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.

[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 and 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 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.

[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.

[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.

[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 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.

[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.

[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.

[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 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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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. 

[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.

[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.

[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 and 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 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.

[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.

[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.

[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 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.

[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.

[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.

[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 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.

[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.

[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.

[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.

[1]

Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure and Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
[2]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
[3]

Pablo G. Barrientos, Artem Raibekas. Robustly non-hyperbolic transitive symplectic dynamics. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 5993-6013. doi: 10.3934/dcds.2018259
[4]

Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61
[5]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[6]

Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[7]

Xiaolong Li, Katsutoshi Shinohara. On super-exponential divergence of periodic points for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1707-1729. doi: 10.3934/dcds.2021169
[8]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
[9]

B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure and Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
[10]

Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545
[11]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[12]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[13]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002
[14]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911
[15]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
[16]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004
[17]

Daniel Franco, Juan Perán, Juan Segura. Stability for one-dimensional discrete dynamical systems revisited. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 635-650. doi: 10.3934/dcdsb.2019258
[18]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[19]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[20]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (156)
  • HTML views (106)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy