Multiple Nonlinear Oscillations in a 𝔻₃×𝔻₃-Symmetrical Coupled System of Identical Cells with Delays

Table 1

The maximal isotropy subgroups of Γ×𝑆¹ and associated fixed point subspaces.

No.	Σ	Fix(Σ,𝑆𝑃𝜔)	dimFix(Σ,𝑆𝑃𝜔)
1	⟨(𝜅,1,1),(1,𝜅,1)⟩	(𝑦,𝑧,𝑦,𝑧,𝑦,𝑧,𝑦,𝑧)	2
2	⟨(𝜅,1,1),(1,𝜅,−1)⟩	(𝑦,𝑧,−𝑦,−𝑧,𝑦,𝑧,−𝑦,−𝑧)	2
3	⟨(𝜅,1,−1),(1,𝜅,1)⟩	(𝑦,𝑧,𝑦,𝑧,−𝑦,−𝑧,−𝑦,−𝑧)	2
4	⟨(𝜅,1,−1),(1,𝜅,−1)⟩	(𝑦,𝑧,−𝑦,−𝑧,−𝑦,−𝑧,𝑦,𝑧)	2
5	⟨(𝜌,1,𝑒^−2𝜋𝑖/3),(1,𝜌,𝑒^−2𝜋𝑖/3)⟩	(𝑦,𝑧,0,0,0,0,0,0)	2
6	⟨(𝜌,1,𝑒^−2𝜋𝑖/3),(1,𝜌,𝑒^2𝜋𝑖/3)⟩	(0,0,𝑦,𝑧,0,0,0,0)	2
7	⟨(𝜌,1,𝑒^2𝜋𝑖/3),(1,𝜌,𝑒^−2𝜋𝑖/3)⟩	(0,0,0,0,𝑦,𝑧,0,0)	2
8	⟨(𝜌,1,𝑒^2𝜋𝑖/3),(1,𝜌,𝑒^2𝜋𝑖/3)⟩	(0,0,0,0,0,0,𝑦,𝑧)	2
9	⟨(𝜌,1,𝑒^−2𝜋𝑖/3),(1,𝜅,1)⟩	(𝑦,𝑧,𝑦,𝑧,0,0,0,0)	2
10	⟨(𝜌,1,𝑒^−2𝜋𝑖/3),(1,𝜅,−1)⟩	(𝑦,𝑧,−𝑦,−𝑧,0,0,0,0)	2
11	⟨(𝜌,1,𝑒^2𝜋𝑖/3),(1,𝜅,1)⟩	(0,0,0,0,𝑦,𝑧,𝑦,𝑧)	2
12	⟨(𝜌,1,𝑒^2𝜋𝑖/3),(1,𝜅,−1)⟩	(0,0,0,0,𝑦,𝑧,−𝑦,−𝑧)	2
13	⟨(1,𝜌,𝑒^−2𝜋𝑖/3),(𝜅,1,1)⟩	(𝑦,𝑧,0,0,𝑦,𝑧,0,0)	2
14	⟨(1,𝜌,𝑒^−2𝜋𝑖/3),(𝜅,1,−1)⟩	(𝑦,𝑧,0,0,−𝑦,−𝑧,0,0)	2
15	⟨(1,𝜌,𝑒^2𝜋𝑖/3),(𝜅,1,1)⟩	(0,0,𝑦,𝑧,0,0,𝑦,𝑧)	2
16	⟨(1,𝜌,𝑒^2𝜋𝑖/3),(𝜅,1,−1)⟩	(0,0,𝑦,𝑧,0,0,−𝑦,−𝑧)	2
17	⟨(𝜌,𝜌,1),(𝜅,𝜅,1)⟩	(0,0,𝑦,𝑧,𝑦,𝑧,0,0)	2
18	⟨(𝜌,𝜌,1),(𝜅,𝜅,−1)⟩	(0,0,𝑦,𝑧,−𝑦,−𝑧,0,0)	2
19	⟨(𝜌^2,𝜌,1),(𝜅,𝜅,1)⟩	(𝑦,𝑧,0,0,0,0,𝑦,𝑧)	2
20	⟨(𝜌^2,𝜌,1),(𝜅,𝜅,−1)⟩	(𝑦,𝑧,0,0,0,0,−𝑦,−𝑧)	2