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A robustly transitive diffeomorphism of Kan's type
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 |
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.
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. |
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