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The generalised version of a theory describing synchronisation in an ensemble shows that coherence arises differently depending on whether the number of dimensions is even or odd. But how could you extend the Kuramoto model to arbitrary dimensions?

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  • $\begingroup$ The original Kuramoto model where each of the oscillators ($i=1 \cdots n$) phase ($\theta_{i}^{}$) coupled to other oscillators phases in the same way i.e., $\dot\theta_{i}^{}=\omega_{i}^{}+\frac{g}{n}\sum_{j=1}^{n}\sin[\theta_{i}^{}-\theta_{j}^{}]$ doesn't carry any information of dimensionality as it should be clear. Did you mean some variant of form for example, $\dot\theta_{i}^{}=\omega_{i}^{}+\sum_{\text{conditional}(j,i)}^{}g_{ij}^{}\sin[\theta_{i}^{}-\theta_{j}^{}]$ defined on some lattice (which certainly has dimensionality aspect). $\endgroup$ – Sunyam Jan 10 at 14:17
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I read an article on it. I will show about it.

Devised in the 1970s to describe synchronization among interacting oscillators, the Kuramoto model has become critical for understanding complex dynamics in systems of interacting entities, such as magnetic dipoles or neurons. Though it has been applied to many disparate fields, the classical version is restricted to interactions that can be described with just two dimensions. By extending the model to three dimensions and more, and by accounting for variation in the tendency of each agent to remain independent, Sarthak Chandra and colleagues at the University of Maryland, College Park, have discovered an unexpected result: The way in which a system achieves coherence—a measure of the degree of synchronization—depends on whether its number of dimensions is even or odd.

In the classical Kuramoto model, the transition to coherence is continuous and begins when the coupling strength between individual components reaches a certain positive threshold. Its 3D generalization, in contrast, shows a sudden, discontinuous transition as soon as the coupling strength rises above zero. When the team extended the model to higher dimensions, they found that all even-dimensional systems show the same behavior as the classical case, while the discontinuous transition occurs for all odd-dimensional systems.

Despite the application of the Kuramoto model to systems as diverse as neural networks, magnetic spins, and consensus-building within societies, its restriction to two dimensions meant that it couldn’t describe some interesting phenomena. By extending the model, Chandra and colleagues have made it more relevant to realistic cases of flocking behavior, like that of animals or drones arranging themselves in 3D space. Future developments will include more variables characterizing the relationship between a given individual and its neighbors, such as distance or relative orientation.

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