The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本 由紀 Kuramoto Yoshiki), is a mathematical model used to describe synchronization. It is a model for the behavior of a large set of coupled phase oscillators. The model is often adopted to study synchronization phenomena in large populations of interacting elements in diverse fields such as physics, biology, chemistry and social systems. The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects (Wikipedia).

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**Kuramoto Model**

**IMPLEMENTATION 1 - ELM**

The hopf oscillator was first implemented in ELM to study the grey-scale synchronization phenomena between adjacent pixels. Different syhnchronization patterns can be observed in the web installation "Contingency and Syhnchronization".

**IMPLEMENTATION 2 - SC UGEN**

The Hopf Ugen was the basis to create mechanisms of frequency avoidance (example), or frequency synchronization, that were used both in the "Contingency and Synchronization" sound installation and in our series of jam sessions.

We were interested in understanding under which conditions, and to what extent, the dynamics of this model could emerge as a perceptible quality of aural - or visual - form.

These experiments conrcetised in a sound installation, a web installation and a series of electronic jam sessions that explore the phenomenon of synchronization.

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**HOPF OSCILLATOR**

The model was implemented through a set of adaptive frequency hopf oscillators. An hopf oscillator is an oscillator that can adapt its parameters to learn the frequency of any periodic input. It means that it changes its parameters in order to have an intrinsic frequency that corresponds to the frequency of the input. This mechanism goes beyond mere synchronization since the new frequency stays encoded in the system, even if the teaching signal disappears and it works for any initial conditions. When hopf oscillators are coupled with each other they form a dynamical system, that generates different behaviors over time. Both local and global synchronization phenomena can be observed and the rate of variation is directly dependent on the coupling strength.

↗ implemented in ELM

↗ implemented as SC Ugen

[link dp, online meeting with hhr, ld 16-jun-2019]

https://biorob.epfl.ch/research/research-dynamical/page-36365-en-html/

21st March 2019

DP

Synchronization is an umbrella word that comes from non-linear dynamics. In synchronization we look at oscillators. Oscillators are processes that repeat periodically, regularly and produce recurring events. An example is the metronome which has always the same period and events always at the same time. Synchronization is the study of what happens if you take an oscillator and expose it to the world, and see how the world interacts with the oscillator, and how the oscillator reacts to it, the world being in mathematical models, some disturbance. Disturbance can happen in different forms, being other oscillators or noise or whatever you put your oscillator in contact with. So it's a study of open systems, which is a very interesting thing in mathematics and dynamical systems, because you usually tend to close your systems in order to analyze them better. [...] We try to elucidate the aesthetically aspects of this behavior.

LD

Generally we are interested in how form emerges through relations. Form is not something preexisting a process, but is something that emerges through connections. That also has some important repercussions for the idea of simulation, because simulation - at least understood in a certain way - assumes that a form is already given, and you find a way to produce or to reproduce a form. We are not interested in that type of simulation, in that representational sense, but much rather in taking also computational processes as something material that can connect, that can relate to another type of reality or to other types of form, and thereby give rise again to new orders of magnitude or to new forms and to new ideas.

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