Source
Stewart, Golubitsky and Pivato (2003), LIT-1674, §§7–9 (pp. 632–644 of SIAM J. Appl. Dyn. Syst. 2(4), read from the accepted version); read in NOTE-1217. It rests on the balanced relations of THEORY-1065.
What was actually shown
The quotient. For a balanced relation ⋈, the natural quotient network has the ⋈-classes as cells, the projections of arrows not joining two distinct equivalent cells as arrows, and two arrows of the same type when an input isomorphism carries the cells of one colour onto those of the other (construction (A)–(F) and eqs. 7.3–7.4, pp. 634–636; Lemma 7.3). It is a quotient map in the sense of Definition 8.1 (Theorem 8.3, p. 639), and any quotient map induces a balanced relation (Lemma 8.2(c), p. 638), so quotients and balanced relations are the same patterns seen two ways.
The theorem. “For any f ∈ F_{G₁}^P, the induced vector field f̄ lies in F_{G₂}^P̄” (Theorem 9.2, p. 642): restricting an admissible field to the polydiagonal and reading it in the quotient’s coordinates (eq. 9.3) gives a field admissible for the quotient. Hence “interesting dynamics (rotating waves, symmetric chaos) in this subspace for the cell system G₂ corresponds to the same dynamics in the cell system G₁, in which ⋈-equivalent cells are synchronous” (p. 642).
Symmetry in the quotient only. The five-cell network of Example 7.1 restricts on {(x, y, x, y, z)} to ẋ = f(x, y, z), ẏ = f(y, z, x), ż = f(z, x, y) (eq. 7.2), the general D₃-equivariant field: “So it is possible for a quotient network to have symmetry even when the original network has none” (p. 633). The three-cell master–slave network of Figure 19 has a quotient with Z₂ symmetry (p. 635), and the seven-cell chain with a back connection has the three-cell ring as quotient, so the ring’s rotating wave is a travelling wave of the chain (eqs. 1.8–1.9, Figure 8, pp. 614–615).
What could have come out otherwise, and did. The quotient’s admissible class is larger than the restrictions: in the four-cell network of Figure 20, restricted fields have linear terms αu₁ + β(u₂ + 2u₃) where a general quotient field has αu₁ + βu₂ + γu₃, so “vector fields that are admissible with respect to the quotient network G_⋈ do not always lift” (Remark 7.5, pp. 637–638), “a groupoid analogue of hidden symmetry”. And quotients with the same relation need not be isomorphic (Example 8.5, Figure 21); the natural one is universal, every other factoring through it (Theorem 8.4, p. 639) as an edge-refinement (Corollary 8.8, p. 641).
What this does not say
- That a quotient solution is stable in the original network. The quotient describes motion within the synchronous subspace. Stability transverse to it, which decides whether the synchrony is seen, is not treated; the paper studies “existence, not stability” (p. 626).
- That every generic behaviour of the quotient occurs. Restriction yields a subclass of the quotient’s admissible fields (Remark 7.5), and φ̂ “is never surjective when φ is not the natural quotient map” (p. 644). The full criterion for lifting is deferred to Dias and Stewart (ref. [5]), unread.
- That the quotient is a coarse-graining of every state. It describes only states in the synchronous subspace; off it the original network’s dynamics are not reduced.
- That the symmetric dynamics are proved here. Rotating waves, multirhythms and symmetric chaos in the examples are shown by simulation (Figures 8, 16–18) and by citing symmetric Hopf bifurcation and the H/K theorem from the authors’ earlier work.