The coupled cell systems for the networks in the slide $183$ have the form:

$\{\begin{array}{ccc}\text{:}{x}_1^{}=f(x_1\frac{,}{b}ar& (x_3,x_4),& )\end{array}$

$\{\begin{array}{ccc}\text{:}{x}_2^{}=f(x_2\frac{,}{b}ar& (x_1,x_3),& )\end{array}$

$\{\begin{array}{ccc}\text{:}{x}_3^{}=f(x_3\frac{,}{b}ar& (x_1,x_4),& )\end{array}$

$\{\begin{array}{ccc}\text{:}{x}_4^{}=f(x_4\frac{,}{b}ar& (x_1,x_3),& )\end{array}$

Assume $f$:$R^{3}RR.$ Show that:

$($a$)$ For both networks, the space

${}_1=\{x$:$x_2=x_4\}$

is flow$-$invariant, that is$,{}_1$ is a network synchrony space.

$($b$)$ The coupled cell systems restricted to ${}_1$ correspond to the three$-$cell $S_3-$symmetric network.

$($c$)$ When the eigenvalue $f_u(0)-f_v(0)$ is critical and $f_u(0)+2f_v(0)0,f_u(0)0,$ then the critical space $E^c$ for the

coupled cell systems associated to both $4-$cell networks satisfies

$E^c$sube${}_1$