After the seminal works by Schelling, several authors have considered models representing binary choices by different kinds of agents or groups of people. The role of the memory in these models is still an open research argument, on which scholars are investigating. The dynamics of binary choices with impulsive agents has been represented, in the recent literature, by a one-dimensional piecewise smooth map. Following a similar way of modeling, we assume a memory effect which leads the next output to depend on the present and the last state. This results in a two-dimensional piecewise smooth map with a limiting case given by a piecewise linear discontinuous map, whose dynamics and bifurcations are investigated. The map has a particular structure, leading to trajectories belonging only to a pair of straight lines. The system can have, in general, only attracting cycles, but the related periods and periodicity regions are organized in a complex structure of the parameter space. We show that the period adding structure, characteristic for the one-dimensional case, also persists in the two-dimensional one. The considered cycles have a symbolic sequence which is obtained by the concatenation of the symbolic sequences of cycles, which play the role of basic cycles in the bifurcation structure. Moreover, differently from the one-dimensional case, the coexistence of two attracting cycles is now possible. The bistability regions in the parameter space are investigated, evidencing the role of different kinds of codimension-two bifurcation points, as well as in the phase space and the related basins of attraction are described.
REFERENCES
1.
W. B.
Arthur
, “Inductive reasoning and bounded rationality
,” Am. Econ. Rev.
84
, 406
–411
(1994
).2.
V.
Avrutin
, M.
Schanz
, and S.
Banerjee
, “Multi-parametric bifurcations in a piecewise-linear discontinuous map
,” Nonlinearity
19
, 1875
–906
(2006
). 3.
V.
Avrutin
, M.
Schanz
, and L.
Gardini
, “Calculation of bifurcation curves by map replacement
,” Int. J. Bifurc. Chaos
20
, 3105
–3135
(2010
). 4.
V.
Avrutin
, M.
Schanz
, and S.
Banerjee
, “Occurrence of multiple attractor bifurcations in the two-dimensional piecewise linear normal form map
,” Nonlinear Dyn.
67
, 293
–307
(2012
). 5.
V.
Avrutin
, L.
Gardini
, I.
Sushko
, and F.
Tramontana
, Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps
(World Scientific Publishing
, 2019
).6.
N.
Bakhshani
, “Impulsivity: A predisposition toward risky behaviors
,” Int. J. High Risk Behav. Addict.
3
(2
), e20428
(2014
). 7.
G. I.
Bischi
and U.
Merlone
, “Evolutionary minority games with memory
,” J. Evol. Econ.
27
(5
), 859
–875
(2017
). 8.
G. I.
Bischi
, U.
Merlone
, and E.
Pruscini
, “Evolutionary dynamics in club goods binary games
,” J. Econ. Dyn. Control
91
, 104
–119
(2018
). 9.
G. I.
Bischi
and U.
Merlone
, “Global dynamics in binary choice models with social influence
,” J. Math. Sociol.
33
(4
), 277
–302
(2009
). 10.
G. I.
Bischi
, L.
Gardini
, and U.
Merlone
, “Impulsivity in binary choices and the emergence of periodicity
,” Disc. Dyn. Nat. Soc.
407913
(2009
). 11.
A.
Cavagna
, “Irrelevance of memory in the minority game
,” Phys. Rev. E
59
(4
), 3783
–3786
(1999
). 12.
D.
Challet
and M.
Marsili
, “Relevance of memory in minority games
,” Phys. Rev. E
62
(2
), 1862
–1868
(2000
). 13.
W.
Chen
, S.-Y.
Liu
, C.-H.
Chen
, and Y.-S.
Lee
, “Bounded memory, inertia, sampling and weighting model for market entry games
,” Games
2
, 187
–199
(2011
). 14.
E.
Ciaramelli
, F.
Bernardi
, and M.
Moscovitch
, “Individualized Theory of Mind (iToM): When memory modulates empathy
,” Front. Psychol.
4
, 1
–4
(2013
). 15.
A.
Dal Forno
and U.
Merlone
, “Border-collision bifurcations in a model of Braess paradox
,” Math. Comput. Simul.
87
, 1
–18
(2013
). 16.
F.
Dellu-Hagedorn
, “Relationship between impulsivity, hyperactivity and working memory: A differential analysis in the rat
,” Behav. Brain. Funct.
2
(1
), 10
(2006
). 17.
P.
Dindo
, “A tractable evolutionary model for the minority game with asymmetric payoffs
,” Physica A
35
, 110
–118
(2005
). 18.
M.
di Bernardo
, C. J.
Budd
, A. R.
Champneys
, and P.
Kowalczyk
, Piecewise-Smooth Dynamical Systems: Theory and Applications
(Springer
, 2008
).19.
R. I. M.
Dunbar
, “The social brain: Mind, language, and society in evolutionary perspective
,” Annu. Rev. Anthropol.
32
(1
), 163
–181
(2003
). 20.
P. S.
Dutta
, B.
Routroy
, S.
Banerjee
, and S. S.
Alam
, “On the existence of low-period orbits in n-dimensional piecewise linear discontinuous maps
,” Nonlinear Dyn.
53
, 369
–380
(2008
). 21.
P. S.
Dutta
and S.
Banerjee
, “Period increment cascades in a discontinuous map with square-root singularity
,” Discr. Contin. Dyn. Syst. Ser. B
14
, 961
–976
(2010
). 22.
A.
Galstyan
, S.
Kolar
, and K.
Lerman
, “Resource Allocation Games with Changing Resource Capacities,” in Proceedings of the Second International Joint Conference on Autonomous Agents and Multiagent Systems (ACM, New York, 2003), pp. 145–152.23.
L.
Gardini
, F.
Tramontana
, V.
Avrutin
, and M.
Schanz
, “Border collision bifurcations in 1D PWL map and Leonov’s approach
,” Int. J. Bifurc. Chaos
20
(10
), 3085
–3104
(2010
). 24.
L.
Gardini
, I.
Sushko
, and K.
Matsuyama
, “2D discontinuous piecewise linear map: Emergence of fashion cycles
,” Chaos
28
, 1
–20
(2018
). 25.
P.
Glendinning
, “Robust chaos revisited
,” Eur. Phys. J. Spec. Top.
226
, 1721
–1738
(2017
). 26.
M.
Granovetter
, “Threshold models of collective behavior
,” Am. J. Sociol.
83
(1
), 1420
–1443
(1978
). 27.
M.
Granovetter
and R.
Soong
, “Threshold models of diffusion and collective behavior
,” J. Math. Sociol.
9
(3
), 165
–179
(1983
). 28.
A. J.
Homburg
, Global Aspects of Homoclinic Bifurcations of Vector Fields, Memoires of the American Mathematical Society
(American Mathematical Society
, 1996
), pp. 1
–68
.29.
C.
Hommes
, T.
Kiseleva
, Y.
Kuznetsov
, and M.
Verbic
, “Is more memory in evolutionary selection (de)stabilizing?
,” Macroecon. Dyn.
16
(3
), 335
–357
(2012
). 30.
A. S.
James
, S. M.
Groman
, E.
Seu
, M.
Jorgensen
, L. A.
Fairbanks
, and J. D.
Jentsch
, “Dimensions of impulsivity are associated with poor spatial working memory performance in monkeys
,” J. Neurosci.
27
(52
), 14358
–14364
(2007
). 31.
J. P.
Keener
, “Chaotic behavior in piecewise continuous difference equations
,” Trans. Am. Math. Soc.
261
(2
), 589
–604
(1980
). 32.
L. E.
Kollar
, G.
Stepan
, and J.
Turi
, “Dynamics of piecewise linear discontinuous maps
,” Int. J. Bifurc. Chaos
14
, 2341
–2351
(2004
). 33.
L.-F.
Lee
, “Identification and estimation in binary choice models with limited (censored) dependent variables
,” Econometrica
4
(47
), 977
–996
(1979
). 34.
N. N.
Leonov
, “On a discontinuous piecewise-linear pointwise mapping of a line into itself
,” Radiofisika
3
(3
), 496
–510
(1960
) (in Russian).35.
N. N.
Leonov
, “On the theory of a discontinuous mapping of a line into itself
,” Radiofisika
3
(5
), 872
–886
(1960
) (in Russian).36.
C.
Mira
, “Embedding of a dim1 piecewise continuous and linear Leonov map into a dim2 invertible map,” in Global Analysis of Dynamic Models for Economics, Finance and Social Sciences, edited by G. I. Bischi, C. Chiarella, and I. Sushko (Springer, 2013), pp. 337–367.37.
C.
Mira
, L.
Gardini
, A.
Barugola
, and J.-C.
Cathala
, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific Series on Nonlinear Science (World Scientific, 1996), Vol. 20.38.
D.
Mookherjee
and B.
Sopher
, “Learning behavior in an experimental matching pennies game
,” Games Econ. Behav.
7
(1
), 62
–91
(1994
). 39.
E.
Moro
, “The Minority Game: An introductory guide,” in Advances in Condensed Matter and Statistical Physics, edited by E. Korutcheva, and R. Cuerno (Nova Science Publishers, 2004), pp. 263–286.40.
A.
Naimzada
and M.
Pireddu
, “Fashion cycle dynamics in a model with endogenous discrete evolution of heterogeneous preferences
,” Chaos
28
, 055907
(2018
). 41.
D. B.
Neill
, “Optimality under noise: Higher memory strategies for the alternating prisoner’s dilemma
,” J. Theor. Biol.
211
(2
), 159
–180
(2001
). 42.
H. E.
Nusse
and J. A.
Yorke
, “Border-collision bifurcations including period two to period three for piecewise smooth systems
,” Physica D
57
, 39
–57
(1992
). 43.
H. E.
Nusse
and J. A.
Yorke
, “Border-collision bifurcations for piecewise smooth one-dimensional maps
,” Int. J. Bifurc. Chaos
5
(1
), 189
–207
(1995
). 44.
J. H.
Patton
, M. S.
Stanford
, and E. S.
Barratt
, “Factor structure of the Barratt impulsiveness scale
,” J. Clin. Psychol.
51
(6
), 768
–774
(1995
). 45.
B.
Rakshit
, M.
Apratim
, and S.
Banerjee
, “Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps
,” Chaos
20
, 033101
(2010
). 46.
A. D.
Redish
and S. J. Y.
Mizumori
, “Memory and decision making
,” Neurobiol. Learn. Mem.
117
, 1
–3
(2015
). 47.
T. C.
Schelling
, “Hockey helmets, concealed weapons, and daylight saving
,” J. Conflict Resolut.
17
, 381
–428
(1973
). 48.
D.
Shohamy
and N. D.
Daw
, “Integrating memories to guide decisions
,” Curr. Opin. Behav. Sci.
5
, 85
–90
(2015
). 49.
D. J. W.
Simpson
, Bifurcations in Piecewise-Smooth Continuous Systems
(World Scientific
, 2010
).50.
D. J. W.
Simpson
, see https://arxiv.org/abs/1907.02653v1 for “Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps” (2019).51.
M.
Sysi-Aho
, A.
Chakraborti
, and K.
Kaski
, “Searching for good strategies in adaptive minority games
,” Phys. Rev. E
69
(3
), 036125
(2004
). 52.
I.
Sushko
, F.
Tramontana
, F.
Westerhoof
, and V.
Avrutin
, “Symmetry breaking in a bull and bear financial market model
,” Chaos Solitons Fractals
79
, 57
–72
(2015
). 53.
F.
Tramontana
, L.
Gardini
, V.
Avrutin
, and M.
Schanz
, “Period adding in piecewise linear maps with two discontinuities
,” Int. J. Bifurc. Chaos
22
(3
), 1250068
(2012
). 54.
Z. T.
Zhusubaliyev
and E.
Mosekilde
, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems
(World Scientific
, 2003
).© 2019 Author(s).
2019
Author(s)
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