Q:
The coupled cell systems for the networks
have the form
{(:x_(1)^(Ë™)=f(x_(1)(,)/(b)ar (x_(3),x_(4)),lambda )):}
{(:x_(2)^(Ë™)=f(x_(2)(,)/(b)ar (x_(1),x_(3)),lambda )):}
{(:x_(3)^(Ë™)=f(x_(3)(,)/(b)ar (x_(1),x_(4)),lambda )):}
{(:x_(4)^(Ë™)=f(x_(4)(,)/(b)ar (x_(1),x_(3)),lambda )):}
Assume f:R^(3) imes R->R, Show that:
(a) For both networks, the space
Delta _(1)={x:x_(2)=x_(4)}
is flow-invariant, that is, Delta _(1) is a network synchrony space.
(b) The coupled cell systems restricted to Delta _(1) correspond to the three-cell
S_(3)-5ymmetric 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)subeDelta _(1)
Q: The coupled cell systems for the networks
have the form
=fx13x4,X =f(x2,3,X =fx14,X 4=fx4,x1,x3,X
=f,x3,4,X 2=fx213,X 3=fx3,x2,z1,X) 4=f4,x1,x3,X
Assume f : R3 R R, Show that:
(a) For both networks, the space
={X :2 =x4}
is flow-invariant, that is, , is a network synchrony space.
(b) The coupled cell systems restricted to , correspond to the three-cell S3-5ymmetric network.
(c When the eigenvalue fu0 f0 is critical and fu0 + 2fu0 0, fu(0) 0, then the critical space Ec for the coupled cell systems associated to both 4 -cell networks satisfies
Ec C .
