. 2019 Jan 23;29(1):011106. doi: 10.1063/1.5083621

TABLE I.

Categorization of the synchronous states: the name of the state is given as $S n_{x}$ : $n$ is the major category index and $x$ is composed of $d$ , $l^{+}$ , $l^{-}$ , and $l^{0}$ , where $d$ stands for a drifting range of $K$ , $l$ for a locking range of $K$ , $l^{+}$ , $l^{-}$ , $l^{0}$ , respectively, for positive slope, negative slope, and zero slope of the curve $(K_{i}, {ϕ_{i}}^{*})$ in the locking range of $K$ . $Δ \equiv ω - Ω$ . For the details, see the text.

States	Oscillators with $K$ from $K_{min}$ to $K_{max}$	Slope of $(K_{i}, {ϕ_{i}}^{*})$	Sign $(Δ)$	$(R, c_{0})$	Locking range of $K$	Additional condition
$S 1_{l^{0}}$	In-phase synchronous	$0$	$0$	$R > c_{0}$	$[K_{min}, K_{max}]$	$R = 1$
$S 1_{d l^{+}}$	Drifting–locked	$+$	$-$	$R \geq c_{0}$	$\frac{\| Δ \|}{R + c_{0}} < K_{i}$	$K_{min} \leq \frac{\| Δ \|}{R + c_{0}}$
$S 1_{l^{+}}$	Fully locked	$+$	$-$	$R \geq c_{0}$	$[K_{min}, K_{max}]$	$\frac{\| Δ \|}{R + c_{0}} < K_{min}$
$S 2_{d l^{-}}$	Drifting–locked	$-$	$+$	$R > c_{0}$	$\frac{Δ}{R - c_{0}} < K_{i}$	$K_{min} \leq \frac{Δ}{R - c_{0}}$
$S 2_{l^{-}}$	Fully locked	$-$	$+$	$R > c_{0}$	$[K_{min}, K_{max}]$	$\frac{Δ}{R - c_{0}} < K_{min}$
$S 3_{d l^{+} d}$	Drifting–locked–drifting	$+$	$-$	$R < c_{0}$	$\frac{\| Δ \|}{R + c_{0}} < K_{i} < \frac{\| Δ \|}{c_{0} - R}$	$K_{min} \leq \frac{\| Δ \|}{R + c_{0}}$ , $\frac{\| Δ \|}{c_{0} - R} \leq K_{max}$
$S 3_{l^{+} d}$	Locked–drifting	$+$	$-$	$R < c_{0}$	$K_{i} < \frac{\| Δ \|}{c_{0} - R}$	$\frac{\| Δ \|}{R + c_{0}} < K_{min}$ , $\frac{\| Δ \|}{c_{0} - R} \leq K_{max}$
$S 3_{d l^{+}}$	Drifting–locked	$+$	$-$	$R < c_{0}$	$\frac{\| Δ \|}{R + c_{0}} < K_{i}$	$K_{min} \leq \frac{\| Δ \|}{R + c_{0}}$ , $K_{max} < \frac{\| Δ \|}{c_{0} - R}$
$S 3_{l^{+}}$	Fully locked	$+$	$-$	$R < c_{0}$	$[K_{min}, K_{max}]$	$\frac{\| Δ \|}{R + c_{0}} < K_{min}$ , $K_{max} < \frac{\| Δ \|}{c_{0} - R}$
$S 4_{d}$	Fully drifting	None	$+, 0$	$R \leq c_{0}$	None	$\dots$