It is shown that, by appropriate choice of the coupling function \(g,\) any complex clustering pattern can appear as an attractor of the system of \(N\) identical phase oscillators that are coupled identically to each other,
\[ \dot{\theta}_{i}=\omega+\frac{1}{N}\sum_{j=1}^{N}g\left( \theta_{i} -\theta_{j}\right), \]
where \(\theta_{i}\in\mathbb{T=}\left[ 0,2\pi\right) ,\) \(i=1,2,\ldots,N.\) Clustering behavior discussed in the paper cannot be obtained by using simple sinusoidal coupling; it is a purely dynamic phenomenon driven by interactions between the oscillators through the coupling function \(g.\)
First, conditions on \(g\) and its derivatives that ensure the presence of a cluster state with given configuration and given stability are derived. These results are used to design coupling functions that lead to stable clustering with prescribed phase relationships between clusters by representing \(g\) in the form of a Fourier series and choosing the coefficients accordingly. Then, simple harmonic and localized coupling functions that lead to clusters separated equally in phase are studied.
The authors conclude that the coupling function provides a lot of useful information regarding possible dynamics of globally coupled oscillators. It is flexible enough to permit all possible clustering to appear as attracting periodic orbits and can be designed to ensure the existence and stability of specific cluster states. The designed systems may be also capable of performing finite state computation. Open problems are formulated in the final part of the paper.