The Kuramoto Model

1. The Kuramoto Model
Billy Okal
April 12, 2012

2. Introduction The Model Demonstration
Outline
1 Introduction
Oscillators
Synchronization
2 The Model
3 Demonstration
Billy Okal
The Kuramoto Model

3. Introduction The Model Demonstration
Oscillators
Oscillators
Oscillation is a repetitive variation of some measure from a
central value (equilibrium). Can also occur between two of
more states
Most common example is mass attached to a linear spring
when no other forces are allowed to act on it. Equilibrium is
when the spring does not move.
In real life there are external dissipative processes like friction
and resistance which convert some of the energy stored in
oscillators into heat and sound. The leads to damping.
Damped systems can be excited by injecting energy from the
surrounding environment.
Billy Okal
The Kuramoto Model

4. Introduction The Model Demonstration
Synchronization
Synchronization of oscillators
Sometimes oscillators get coupled, meaning certain variables
in one system influences other variables in other systems.
Common example is that of two pendulum clocks of identical
frequency mounted on a common surface
The influence of the coupling might be global or not.
Billy Okal
The Kuramoto Model

5. Introduction The Model Demonstration
Formal Definition
The Kuramoto model makes a number of assumptions, namely;
All oscillators in the system are Globally Coupled.
Oscillators are identical, except for possibly different natural
frequencies
The phase response curve depends on the phase between two
oscillators
The phase response curve is of a sinusoidal form.
Billy Okal
The Kuramoto Model

6. Introduction The Model Demonstration
Formal Definition
Definition (Kuramoto Model)
The Kuramoto model considers s system of globally coupled
oscillators governed by the equation.
dθi K
= ωi + ΣN sin(θj − θi ) (1)
dt N j=1
where is the phase, ω is the natural frequency, N is the number of
oscillators and K is the coupling constant1
1
http:
//www.johnwordsworth.com/tutorials/Kuramoto/index.php?s=4&p=0
Billy Okal
The Kuramoto Model

7. Introduction The Model Demonstration
Demo
Video and Live Demo
Billy Okal
The Kuramoto Model