scholarly journals Equivariant dynamical systems

Scholarpedia ◽  
2007 ◽  
Vol 2 (10) ◽  
pp. 2510 ◽  
Author(s):  
Jeff Moehlis ◽  
Edgar Knobloch
Keyword(s):  
Dynamical Systems ◽  
Equivariant Dynamical Systems

SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 1: FIXED-POINT SPACES

2006 ◽  
Vol 16 (03) ◽  
pp. 559-577 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
IAN STEWART
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Nearest Neighbor ◽  
Invariant Subspaces ◽  
Folk Theorem ◽  
Coupled Cell Networks ◽  
Equivariant Dynamical Systems ◽  
Cell Networks

Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetry-breaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes ("cells"). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

Parameter uncertainties in models of equivariant dynamical systems

1997 ◽  
Vol 102 (3-4) ◽  
pp. 208-226 ◽  
Author(s):  
Reggie Brown ◽  
Visarath In ◽  
Eugene R. Tracy
Keyword(s):  
Dynamical Systems ◽  
Parameter Uncertainties ◽  
Equivariant Dynamical Systems
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