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 [
Citation: |
Table 1. Domains of continuity contained in Cube 1.
Table 2. Domains of continuity contained in Cube 2.
Table 3. Domains of continuity contained in Cube 3.
Table 4. Domains of continuity contained in Cube 4.
Table 5. Domains of continuity contained in Cube 5.
Table 6. Domains of continuity contained in Cube 6.
Table 7. Domains of continuity contained in Cubes 7 and 8.
Table 8.
Values of
0 | 2 | 0 | 1 | 1 | 0 | 1 | 4 | 3 | 2 | 4 | 3 | 4 | 0 | 1 | 2 | 0 | 1 | 0 | 4 | 2 | 4 | 3 | 3 | 4 | 3 | |||||||||
0 | 1 | 1 | 0 | 2 | 4 | 4 | 4 | 3 | 3 | 0 | 1 | 1 | 0 | 1 | 1 | 4 | 3 | 3 | 4 | 3 | 3 | 4 | 2 | 0 | 0 | |||||||||
0 | 0 | 2 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 2 | 4 | 3 | 4 | 4 | 3 | 2 | 4 | 4 | 2 | 3 | 3 | 4 | 3 |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Chazottes and B. Fernandez, Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, volume 671, Springer Science & Business Media, 2005.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
E. Järvenpää, A SRB-measure for globally coupled circle maps, Nonlinearity, 6 (1997), 1435.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. Koiller
and L. S. Young
, Coupled map networks, Nonlinearity, 23 (2010)
, 1121-1141.
doi: 10.1088/0951-7715/23/5/006.![]() ![]() ![]() |
|
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.![]() ![]() |
|
W. Parry, The Lorenz attractor and a related population model, in Ergodic Theory, pages 169–187, Lecture Notes in Math., 729, Springer, Berlin, 1979.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
D. Thomine, A spectral gap for transer operators of piecewise expanding maps, Discrete Contin. Dyn. Syst., 30 (2011), 917–944, arXiv: 1006.2608.
![]() ![]() |
The function
The graph of
The asymmetric set
Six asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries as according to the legend.
Eight conjectured asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries according to the legend.
The symmetric set
Overview of the critical parameters and the corresponding invariant sets. Results are marked with black, conjectures with gray.
Continuity domains of the map
Cubes
Cubes
Cubes
Cubes 7 and 8.
The image of
The sets
The sets
Images of