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Symmetry breaking in a globally coupled map of four sites
1. | Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary |
2. | MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Egry József u. 1, H-1111 Budapest, Hungary |
A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites [
References:
[1] |
C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti,
Ising-type transitions in coupled map lattices, Journal of Statistical Physics, 80 (1995), 1185-1205.
doi: 10.1007/BF02179868. |
[2] |
C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti,
Ising-type and other transitions in one-dimensional coupled map lattices with sign symmetry, Journal of Statistical Physics, 102 (2001), 1271-1283.
doi: 10.1023/A:1004892312745. |
[3] |
L. A. Bunimovich and Y. G. Sinai,
Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1998), 491-516.
doi: 10.1088/0951-7715/1/4/001. |
[4] |
J. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, volume 671, Springer Science & Business Media, 2005. |
[5] |
B. Fernandez,
Breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps, Journal of Statistical Physics, 154 (2014), 999-1029.
doi: 10.1007/s10955-013-0903-9. |
[6] |
G. Gielis and R. S. MacKay,
Coupled map lattices with phase transition, Nonlinearity, 13 (2000), 867-888.
doi: 10.1088/0951-7715/13/3/320. |
[7] |
E. Järvenpää, A SRB-measure for globally coupled circle maps, Nonlinearity, 6 (1997), 1435. |
[8] |
M. Jiang and Y.B. Pesin,
Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Communications in Mathematical Physics, 193 (1998), 675-711.
doi: 10.1007/s002200050344. |
[9] |
W. Just,
Globally coupled maps: Phase transitions and synchronization, Physica D: Nonlinear Phenomena, 81 (1995), 317-340.
doi: 10.1016/0167-2789(94)00213-A. |
[10] |
G. Keller and C. Liverani,
Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Communications in Mathematical Physics, 262 (2006), 33-50.
doi: 10.1007/s00220-005-1474-7. |
[11] |
J. Koiller and L. S. Young,
Coupled map networks, Nonlinearity, 23 (2010), 1121-1141.
doi: 10.1088/0951-7715/23/5/006. |
[12] |
J. Miller and D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Physical Review E, 48 (1993), 2528.
doi: 10.1103/PhysRevE.48.2528. |
[13] |
W. Parry, The Lorenz attractor and a related population model, in Ergodic Theory, pages 169–187, Lecture Notes in Math., 729, Springer, Berlin, 1979. |
[14] |
F. Sélley and P. Bálint,
Mean-field coupling of identical expanding circle maps, Journal of Statistical Physics, 164 (2016), 858-889.
doi: 10.1007/s10955-016-1568-y. |
[15] |
D. Thomine, A spectral gap for transer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917–944, arXiv: 1006.2608. |
show all references
References:
[1] |
C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti,
Ising-type transitions in coupled map lattices, Journal of Statistical Physics, 80 (1995), 1185-1205.
doi: 10.1007/BF02179868. |
[2] |
C. Boldrighini, L. A. Bunimovich, G. Cosimi, S. Frigio and A. Pellegrinotti,
Ising-type and other transitions in one-dimensional coupled map lattices with sign symmetry, Journal of Statistical Physics, 102 (2001), 1271-1283.
doi: 10.1023/A:1004892312745. |
[3] |
L. A. Bunimovich and Y. G. Sinai,
Spacetime chaos in coupled map lattices, Nonlinearity, 1 (1998), 491-516.
doi: 10.1088/0951-7715/1/4/001. |
[4] |
J. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, volume 671, Springer Science & Business Media, 2005. |
[5] |
B. Fernandez,
Breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps, Journal of Statistical Physics, 154 (2014), 999-1029.
doi: 10.1007/s10955-013-0903-9. |
[6] |
G. Gielis and R. S. MacKay,
Coupled map lattices with phase transition, Nonlinearity, 13 (2000), 867-888.
doi: 10.1088/0951-7715/13/3/320. |
[7] |
E. Järvenpää, A SRB-measure for globally coupled circle maps, Nonlinearity, 6 (1997), 1435. |
[8] |
M. Jiang and Y.B. Pesin,
Equilibrium measures for coupled map lattices: Existence, uniqueness and finite-dimensional approximations, Communications in Mathematical Physics, 193 (1998), 675-711.
doi: 10.1007/s002200050344. |
[9] |
W. Just,
Globally coupled maps: Phase transitions and synchronization, Physica D: Nonlinear Phenomena, 81 (1995), 317-340.
doi: 10.1016/0167-2789(94)00213-A. |
[10] |
G. Keller and C. Liverani,
Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension, Communications in Mathematical Physics, 262 (2006), 33-50.
doi: 10.1007/s00220-005-1474-7. |
[11] |
J. Koiller and L. S. Young,
Coupled map networks, Nonlinearity, 23 (2010), 1121-1141.
doi: 10.1088/0951-7715/23/5/006. |
[12] |
J. Miller and D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice, Physical Review E, 48 (1993), 2528.
doi: 10.1103/PhysRevE.48.2528. |
[13] |
W. Parry, The Lorenz attractor and a related population model, in Ergodic Theory, pages 169–187, Lecture Notes in Math., 729, Springer, Berlin, 1979. |
[14] |
F. Sélley and P. Bálint,
Mean-field coupling of identical expanding circle maps, Journal of Statistical Physics, 164 (2016), 858-889.
doi: 10.1007/s10955-016-1568-y. |
[15] |
D. Thomine, A spectral gap for transer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917–944, arXiv: 1006.2608. |












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