Symmetry breaking in a globally coupled map of four sites

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August 2018, 38(8): 3707-3734. doi: 10.3934/dcds.2018161

Symmetry breaking in a globally coupled map of four sites

Fanni M. Sélley ^1,2,,

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

Received April 2017 Revised March 2018 Published May 2018

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 [14], we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in [14]. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

Keywords: Coupled map systems, ergodicity breaking, asymmetric invariant set, piecewise affine maps, circle maps.

Mathematics Subject Classification: Primary: 37A25, 37E10; Secondary: 37G99.

Citation: Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161

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.

Figure 1. The function $g$.

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Figure 2. The graph of $L_{\varepsilon}$.

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Figure 3. The asymmetric set $\mathcal{A}$.

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Figure 4. Six asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries as according to the legend.

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Figure 5. Eight conjectured asymmetric invariant sets and the symmetries connecting them. Edge colors indicate the symmetries according to the legend.

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Figure 6. The symmetric set $\mathcal{S}$ for $1-\frac{\sqrt{2}}{2} \leq \varepsilon$.

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Figure 7. Overview of the critical parameters and the corresponding invariant sets. Results are marked with black, conjectures with gray.

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Figure 8. Continuity domains of the map $G_{\varepsilon,3}$. $\mathbb{T}^3$ is represented as the unit cube of $\mathbb{R}^3$. The singularities are marked according to the legend.

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Figure 9. Cubes $1$ and $2$.

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Figure 10. Cubes $3$ and $4$.

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Figure 11. Cubes $5$ and $6$.

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Figure 12. Cubes 7 and 8.

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Figure 13. The image of $P_1 \cap 4a$ for $\varepsilon = 0.37 < \varepsilon^*$ and $\varepsilon = 0.41 > \varepsilon^*$.

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Figure 14. The sets $\mathcal{A}$ (black) and $S_0(\mathcal{A})$ (gray). The plane $r-p = 1-2p^*$ separates $S_0(P_1)$, $S_0(P_2)$ and $\mathcal{A}$.

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Figure 15. The sets $\mathcal{A}$ (black) and $S_1(\mathcal{A})$ (gray). Different angles are plotted for better visibility.

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Figure 16. Images of $P_0 \cap 4b$ and $P_0 \cap 1e$ for $\varepsilon = 0.32 > 1-\frac{\sqrt{2}}{2}$.

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Table 1. Domains of continuity contained in Cube 1.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{1a}$	$p > 0$	$q > 0$	$r > 0$			$p+q+r < 1/2$
$\bf{1b}$	$p < 1/2$		$r > 0$	$p+q > 1/2$	$q+r < 1/2$
$\bf{1c}$	$p > 0$		$r < 1/2$	$p+q < 1/2$	$q+r > 1/2$
$\bf{1d}$		$q > 0 $		$p+q < 1/2$	$q+r < 1/2$	$p+q+r > 1/2$
$\bf{1e}$	$p < 1/2$	$q < 1/2 $	$r < 1/2$	$p+q > 1/2$	$q+r > 1/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{1a}$	$p > 0$	$q > 0$	$r > 0$			$p+q+r < 1/2$
$\bf{1b}$	$p < 1/2$		$r > 0$	$p+q > 1/2$	$q+r < 1/2$
$\bf{1c}$	$p > 0$		$r < 1/2$	$p+q < 1/2$	$q+r > 1/2$
$\bf{1d}$		$q > 0 $		$p+q < 1/2$	$q+r < 1/2$	$p+q+r > 1/2$
$\bf{1e}$	$p < 1/2$	$q < 1/2 $	$r < 1/2$	$p+q > 1/2$	$q+r > 1/2$

Table 2. Domains of continuity contained in Cube 2.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{2a}$	$0 < p < 1/2$	$1/2 < q < 1 $	$0 < r < 1/2$			$p+q+r < 3/2$
$\bf{2b}$	$p < 1/2$	$ q < 1 $	$ r < 1/2$			$p+q+r > 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{2a}$	$0 < p < 1/2$	$1/2 < q < 1 $	$0 < r < 1/2$			$p+q+r < 3/2$
$\bf{2b}$	$p < 1/2$	$ q < 1 $	$ r < 1/2$			$p+q+r > 3/2$

Table 3. Domains of continuity contained in Cube 3.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{3a}$	$p >1/2$	$ q > 1/2 $	$ r < 1/2$	$p+q < 3/2$		$p+q+r > 3/2$
$\bf{3b}$	$p > 1/2$	$ q > 1/2 $	$ r > 0$			$p+q+r < 3/2$
$\bf{3c}$	$p < 1$	$ q < 1 $	$ 0 < r < 1/2$	$p+q > 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{3a}$	$p >1/2$	$ q > 1/2 $	$ r < 1/2$	$p+q < 3/2$		$p+q+r > 3/2$
$\bf{3b}$	$p > 1/2$	$ q > 1/2 $	$ r > 0$			$p+q+r < 3/2$
$\bf{3c}$	$p < 1$	$ q < 1 $	$ 0 < r < 1/2$	$p+q > 3/2$

Table 4. Domains of continuity contained in Cube 4.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{4a}$	$1/2 < p < 1$	$ q > 0$	$ r > 0$		$q+r < 1/2$
$\bf{4b}$	$p > 1/2$	$ q < 1/2$	$ r < 1/2$		$q+r > 1/2$	$p+q+r < 3/2$
$\bf{4c}$	$p < 1$	$ q < 1/2$	$ r < 1/2$			$p+q+r > 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{4a}$	$1/2 < p < 1$	$ q > 0$	$ r > 0$		$q+r < 1/2$
$\bf{4b}$	$p > 1/2$	$ q < 1/2$	$ r < 1/2$		$q+r > 1/2$	$p+q+r < 3/2$
$\bf{4c}$	$p < 1$	$ q < 1/2$	$ r < 1/2$			$p+q+r > 3/2$

Table 5. Domains of continuity contained in Cube 5.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{5a}$	$p > 0$	$ q > 0$	$1/2 < r < 1$	$p+q < 1/2$
$\bf{5b}$	$p < 1/2$	$ q < 1/2$	$r > 1/2$	$p+q > 1/2$		$p+q+r < 3/2$
$\bf{5c}$	$p < 1/2$	$ q < 1/2$	$r < 1$			$p+q+r > 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{5a}$	$p > 0$	$ q > 0$	$1/2 < r < 1$	$p+q < 1/2$
$\bf{5b}$	$p < 1/2$	$ q < 1/2$	$r > 1/2$	$p+q > 1/2$		$p+q+r < 3/2$
$\bf{5c}$	$p < 1/2$	$ q < 1/2$	$r < 1$			$p+q+r > 3/2$

Table 6. Domains of continuity contained in Cube 6.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{6a}$	$p < 1/2$	$ q > 1/2$	$r > 1/2$		$q+r < 3/2$	$p+q+r > 3/2$
$\bf{6b}$	$p > 0$	$ q > 1/2$	$r > 1/2$			$p+q+r < 3/2$
$\bf{6c}$	$0 < p < 1/2$	$ q < 1$	$r < 1$		$q+r > 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{6a}$	$p < 1/2$	$ q > 1/2$	$r > 1/2$		$q+r < 3/2$	$p+q+r > 3/2$
$\bf{6b}$	$p > 0$	$ q > 1/2$	$r > 1/2$			$p+q+r < 3/2$
$\bf{6c}$	$0 < p < 1/2$	$ q < 1$	$r < 1$		$q+r > 3/2$

Table 7. Domains of continuity contained in Cubes 7 and 8.

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{7a}$	$p < 1$	$ q < 1$	$r < 1$			$p+q+r > 5/2$
$\bf{7b}$	$p > 1/2$		$r < 1$	$p+q < 3/2$	$q+r > 3/2$
$\bf{7c}$	$p < 1$		$r > 1/2$	$p+q > 3/2$	$q+r < 3/2$
$\bf{7d}$		$q < 1$		$p+q > 3/2$	$q+r > 3/2$	$p+q+r < 5/2$
$\bf{7e}$	$p > 1/2$	$q >1/2$	$r > 1/2$	$p+q < 3/2$	$q+r < 3/2$

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{8a}$	$1/2 < p < 1$	$0 < q < 1$	$1/2 < r < 1$			$p+q+r > 3/2$
$\bf{8b}$	$p > 1/2$	$q >0$	$r > 1/2$			$p+q+r < 3/2$

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	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{7a}$	$p < 1$	$ q < 1$	$r < 1$			$p+q+r > 5/2$
$\bf{7b}$	$p > 1/2$		$r < 1$	$p+q < 3/2$	$q+r > 3/2$
$\bf{7c}$	$p < 1$		$r > 1/2$	$p+q > 3/2$	$q+r < 3/2$
$\bf{7d}$		$q < 1$		$p+q > 3/2$	$q+r > 3/2$	$p+q+r < 5/2$
$\bf{7e}$	$p > 1/2$	$q >1/2$	$r > 1/2$	$p+q < 3/2$	$q+r < 3/2$

	$\bf{p}$	$\bf{q}$	$\bf{r}$	$\bf{p+q}$	$\bf{q+r}$	$\bf{p+q+r}$

$\bf{8a}$	$1/2 < p < 1$	$0 < q < 1$	$1/2 < r < 1$			$p+q+r > 3/2$
$\bf{8b}$	$p > 1/2$	$q >0$	$r > 1/2$			$p+q+r < 3/2$

Table 8. Values of $c_1,c_2$ and $c_3$ in Formula 5, for each continuity domain.

		$\bf{1a}$	$\bf{1b}$	$\bf{1c}$	$\bf{1d}$	$\bf{1e}$		$\bf{2a}$	$\bf{2b}$		$\bf{3a}$	$\bf{3b}$	$\bf{3c}$		$\bf{4a}$	$\bf{4b}$	$\bf{4c}$		$\bf{5a}$	$\bf{5b}$	$\bf{5c}$	$\bf{6a}$	$\bf{6b}$		$\bf{6c}$	$\bf{7a}$		$\bf{7b}$	$\bf{7c}$	$\bf{7d}$		$\bf{7e}$	$\bf{8a}$	$\bf{8b}$

$\bf{c_1}$		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
$\bf{c_2}$		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
$\bf{c_3}$		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

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		$\bf{1a}$	$\bf{1b}$	$\bf{1c}$	$\bf{1d}$	$\bf{1e}$		$\bf{2a}$	$\bf{2b}$		$\bf{3a}$	$\bf{3b}$	$\bf{3c}$		$\bf{4a}$	$\bf{4b}$	$\bf{4c}$		$\bf{5a}$	$\bf{5b}$	$\bf{5c}$	$\bf{6a}$	$\bf{6b}$		$\bf{6c}$	$\bf{7a}$		$\bf{7b}$	$\bf{7c}$	$\bf{7d}$		$\bf{7e}$	$\bf{8a}$	$\bf{8b}$

$\bf{c_1}$		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
$\bf{c_2}$		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
$\bf{c_3}$		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

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