TABLE I.

Categorization of the synchronous states for the case where d₀sinα ≥ 0: The name of the state is given as Sn_x, following the same naming scheme presented in Ref. 16: n is the major category index and x is composed of d, l⁺, l⁻, and l⁰ where d stands for a drifting range of K, l for a locking range of K, and l⁺, l⁻, and l⁰, respectively, for a positive slope, a negative slope, and zero slope of the curve $(K_j,{{\varphi }_j}^{\ast })$ in the locking range of K. Δ ≡ ω − Ω. $D_0\equiv |d_0\sin \alpha |{r_j}^{\ast }$ in the last column with ${r_j}^{\ast }={r_{\text{min}}}^{\ast }$ for K_min and ${r_j}^{\ast }={r_{\text{max}}}^{\ast }$ for K_max. For other details, see the text.

States	Oscillators with K from Kmin to Kmax	Slope of $(K_j,{{\varphi }_j}^{\ast })$	Sign(Δ)	$(\tilde{R},D_0)$	Locking range of K	Additional condition
$S{1}_{l^0}$	In-phase synchronous	0	0	$\tilde{R}>D_0$	[Kmin, Kmax]	$\text{max}\tilde{R},\mathrm{\Delta }=0$
$S{1}_{l^{+}}$	Fully locked	+	−	$\tilde{R}\ge D_0$	[Kmin, Kmax]	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_{\text{min}}$
$S{1}_{d{l}^{+}}$	Drifting–locked	+	−	$\tilde{R}\ge D_0$	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_j$	$K_{\text{min}}\le \frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}$
$S{2}_{l^{-}}$	Fully locked	−	+	$\tilde{R}>D_0$	[Kmin, Kmax]	$\frac{\mathrm{\Delta }{r}_j^{\ast }}{\tilde{R}-D_0}<K_{\text{min}}$
$S{2}_{d{l}^{-}}$	Drifting–locked	−	+	$\tilde{R}>D_0$	$\frac{\mathrm{\Delta }{r}_j^{\ast }}{\tilde{R}-D_0}<K_j$	$K_{\text{min}}\le \frac{\mathrm{\Delta }{r}_j^{\ast }}{\tilde{R}-D_0}$
S2d	Fully drifting	−	+	$\tilde{R}>D_0$	None	$K_{\text{max}}\le \frac{\mathrm{\Delta }{r}_j^{\ast }}{\tilde{R}-D_0}$
$S{3}_{l^{+}}$	Fully locked	+	−	$\tilde{R}<D_0$	[Kmin, Kmax]	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_{\text{min}}$, $K_{\text{max}}<\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}$
$S{3}_{l^{+}d}$	Locked–drifting	+	−	$\tilde{R}<D_0$	$K_j<\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}$	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_{\text{min}}$, $\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}\le K_{\text{max}}$
$S{3}_{d{l}^{+}}$	Drifting–locked	+	−	$\tilde{R}<D_0$	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_j$	$K_{\text{min}}\le \frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}$, $K_{\text{max}}<\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}$
$S{3}_{d{l}^{+}d}$	Drifting–locked–drifting	+	−	$\tilde{R}<D_0$	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}<K_j<\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}$	$K_{\text{min}}\le \frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}$, $\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}\le K_{\text{max}}$
S3d	Fully drifting	None	−	$\tilde{R}<D_0$	None	$\frac{|\mathrm{\Delta }|r_j^{\ast }}{D_0-\tilde{R}}\le K_{\text{min}}$ or $K_{\text{max}}\le \frac{|\mathrm{\Delta }|r_j^{\ast }}{\tilde{R}+D_0}$
S4d	Fully drifting	None	+, 0	$\tilde{R}\le D_0$	None	⋅⋅ ⋅