Assuming that oscillators j and i **are coupled, the gap between their (current) phases is**
defined as:
∆ji = θj − θi (2)
Now, the change in the behavior of oscillator j as influenced by oscillator i **can be described**
as follows:dθj dt = ωj + g(∆ji) , (3)
where g(·**) is the phase shift function which is responsible for changing the phase of the frogs**
that are influenced by other frogs. In [1], the authors suggest the use of the following phase shift function:
g(x) = α sin(x**) (4)**
We say that this system of oscillators is in a stable situation and in anti-phase when the following two conditions are satisfied:

$$\frac{d\theta_{j}}{d t}=\omega_{j}+g(\Delta_{j i})\quad,$$
$\left(3\right)$. 
$$g(x)=\alpha\sin(x)$$
$$\begin{array}{r c l}{{\Delta_{i j}}}&{{=}}&{{\Delta_{j i}\quad,}}\\ {{g(\Delta_{i j})}}&{{=}}&{{0\quad,}}\end{array}$$
$$\begin{array}{l}{(5)}\\ {(6)}\end{array}$$
∆ij = ∆ji , (5)
g(∆ij ) = 0 , (6)
for all i 6= j**. The system presented in [1] is able to successfully locate two coupled oscillators**
in perfect anti-phase, independent of the initial settings of θ1 and θ2**. Unfortunately, several**
problems arise when the number of oscillators grows. Figure **3 shows two examples for such** problems. Given an undirected graph G = (V, E**), henceforth we will assign one oscillator to**
each node in the graph. Therefore, in the following the terms node and oscillator **will refer**
to the same. We consider that two oscillators are coupled if and only if their corresponding nodes are connected by an edge. Depending on the initial phases of the oscillators, for both topologies shown in Figures 3(a) and 3(d) it is possible to reach suboptimal desynchronizations (as shown in Figures 3(b) and 3(e)). The corresponding **optimal desynchronizations are**
shown in Figures 3(c) and 3(f). In [1] the authors provide analytical results for using three oscillators and show that there is a high system sensitivity **with respect to the initial phases** (only a small subset of the possible initial settings leads to an optimal solution).

The initial model by Aihara et al. [1] was later extended by Mutazono et al. [35]. They used their extended model for anti-phase synchronization for the purpose of collision-free transmission scheduling in sensor networks. In order to make the system applicable to larger topologies (sensor networks may consists of hundreds of nodes), they introduced weights in order to regulate the coupling between each pair of oscillators. The resulting phase shift function as introduced in [35] can be described as follows:

$$\begin{array}{r c l}{{\delta(x)}}&{{=}}&{{\operatorname*{min}\{x,2\pi-x\}\quad,}}\\ {{g(x)}}&{{=}}&{{\alpha\sin(x)\cdot e^{-\delta(x)}}}\end{array}$$
−δ(x)(8)
Thanks to these weights, the system reaches stable situations more easily, especially when rather small values of α **are used. The authors experimented with topologies of up to 20**
nodes and although the system still showed certain difficulties to reach stable solutions, the sensitivity to initial conditions decreased significantly.

Mutazono et al. [35] compared the results of their system to another mechanism for coupled oscillator desynchronization proposed in [10]. Note that the mechanism from [10] is not based

$$\begin{array}{l}{(7)}\\ {(8)}\end{array}$$
