Table 1.

Enumeration of all critical pattern spaces, the multiplicity and type of critical eigenvalues, and the dimension of the critical eigenspaces, given certain conditions on the trace and the determinant of the internal and coupled dynamics. The condition “NDG” stands for non-degeneracy and refers to some condition on the trace or determinant of $\mathbf{Q}+{\mu }_i\mathbf{R}$ that ensures it is nonzero for every $i$. An example of such conditions is given in Theorem 10. If ${\beta }_1=1$, the first four bifurcations have simple critical eigenvalues and thus generically lead to a steady-state or Hopf bifurcations with critical pattern spaces $P^{{\mu }_1}$ (synchrony-breaking) or $P^{{\mu }_k}$ (synchrony-preserving).

Critical Pattern Space	Critical Eigenvalues	Dimension of Critical Eigenspace	Determinant Condition	Trace Condition
$P^{{\mu }_1}$	${\alpha }_1$ (real)	${\beta }_1$	$B>0$	NDG
$P^{{\mu }_1}$	$2{\alpha }_1$ (imag.)	${\beta }_1$	NDG	$\text{tr}\left(\mathbf{R}\right)<0$
$P^{{\mu }_k}$	1 (real)	1	$B<0$	NDG
$P^{{\mu }_k}$	2 (imag.)	1	NDG	$\text{tr}\left(\mathbf{R}\right)>0$
$P^{{\mu }_1}$	$2{\alpha }_1$ (real)	$2{\beta }_1$	$B>0$	$\text{tr}\left(\mathbf{R}\right)<0$
$P^{{\mu }_k}$	2 (real)	2	$B<0$	$\text{tr}\left(\mathbf{R}\right)>0$
$P^{{\mu }_1}\oplus P^{{\mu }_k}$	${\alpha }_1$ (real) 2 (imag.)	${\beta }_1+1$	$B>0$	$\text{tr}\left(\mathbf{R}\right)>0$
$P^{{\mu }_1}\oplus P^{{\mu }_k}$	$2{\alpha }_1$ (imag.) 1 (real)	${\beta }_1+1$	$B<0$	$\text{tr}\left(\mathbf{R}\right)<0$
${\mathbb{R}}^n$	$2{\alpha }_1$ (real) $\sum _{j\ne 1}2{\alpha }_j$ (imag.)	$n+{\beta }_1$	$B>0$	$\text{tr}\left(\mathbf{R}\right)=0$
${\mathbb{R}}^n$	2 (real) $\sum _{j\ne k}2{\alpha }_j$ (imag.)	$n+1$	$B<0$	$\text{tr}\left(\mathbf{R}\right)=0$
${\mathbb{R}}^n$	$2{\alpha }_1$ (real) $\sum _{j\ne 1}{\alpha }_j$ (real)	$n+{\beta }_1$	$B=0$	$\text{tr}\left(\mathbf{R}\right)<0$
${\mathbb{R}}^n$	2 (real) $\sum _{j\ne k}{\alpha }_j$ (real)	$n+1$	$B=0$	$\text{tr}\left(\mathbf{R}\right)>0$
${\mathbb{R}}^n$	${\sum }_{1\le j\le k}2{\alpha }_j$ (imag.)	$n$	NDG	$\text{tr}\left(\mathbf{R}\right)=0$
${\mathbb{R}}^n$	${\sum }_{1\le j\le k}{\alpha }_j$ (real)	$n$	$B=0$	NDG
${\mathbb{R}}^n$	${\sum }_{1\le j\le k}2{\alpha }_j$ (real)	$2n$	$B=0$	$\text{tr}\left(\mathbf{R}\right)=0$