Article Contents
Article Contents

Symmetry breaking in a globally coupled map of four sites

• 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.

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

 Citation:

• Figure 1.  The function $g$.

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

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

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

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

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

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

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.

Figure 9.  Cubes $1$ and $2$.

Figure 10.  Cubes $3$ and $4$.

Figure 11.  Cubes $5$ and $6$.

Figure 12.  Cubes 7 and 8.

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

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 $\mathcal{A}$.

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

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

Table 1.  Domains of continuity contained in Cube 1.

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

Table 2.  Domains of continuity contained in Cube 2.

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

Table 3.  Domains of continuity contained in Cube 3.

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

Table 4.  Domains of continuity contained in Cube 4.

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

Table 5.  Domains of continuity contained in Cube 5.

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

Table 6.  Domains of continuity contained in Cube 6.

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

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

 $\bf{p}$ $\bf{q}$ $\bf{r}$ $\bf{p+q}$ $\bf{q+r}$ $\bf{p+q+r}$ $\bf{7a}$ $p < 1$ $q < 1$ $r < 1$ $p+q+r > 5/2$ $\bf{7b}$ $p > 1/2$ $r < 1$ $p+q < 3/2$ $q+r > 3/2$ $\bf{7c}$ $p < 1$ $r > 1/2$ $p+q > 3/2$ $q+r < 3/2$ $\bf{7d}$ $q < 1$ $p+q > 3/2$ $q+r > 3/2$ $p+q+r < 5/2$ $\bf{7e}$ $p > 1/2$ $q >1/2$ $r > 1/2$ $p+q < 3/2$ $q+r < 3/2$ $\bf{p}$ $\bf{q}$ $\bf{r}$ $\bf{p+q}$ $\bf{q+r}$ $\bf{p+q+r}$ $\bf{8a}$ $1/2 < p < 1$ $0 < q < 1$ $1/2 < r < 1$ $p+q+r > 3/2$ $\bf{8b}$ $p > 1/2$ $q >0$ $r > 1/2$ $p+q+r < 3/2$

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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