Kuramoto
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The Kuramoto Model
Transcript of Kuramoto
The Kuramoto Model
Billy Okal
April 12, 2012
Introduction The Model Demonstration
Outline
1 IntroductionOscillatorsSynchronization
2 The Model
3 Demonstration
Billy Okal
The Kuramoto Model
Introduction The Model Demonstration
Oscillators
Oscillators
Oscillation is a repetitive variation of some measure from acentral value (equilibrium). Can also occur between two ofmore states
Most common example is mass attached to a linear springwhen no other forces are allowed to act on it. Equilibrium iswhen the spring does not move.
In real life there are external dissipative processes like frictionand resistance which convert some of the energy stored inoscillators into heat and sound. The leads to damping.
Damped systems can be excited by injecting energy from thesurrounding environment.
Billy Okal
The Kuramoto Model
Introduction The Model Demonstration
Synchronization
Synchronization of oscillators
Sometimes oscillators get coupled, meaning certain variablesin one system influences other variables in other systems.
Common example is that of two pendulum clocks of identicalfrequency mounted on a common surface
The influence of the coupling might be global or not.
Billy Okal
The Kuramoto Model
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 naturalfrequencies
The phase response curve depends on the phase between twooscillators
The phase response curve is of a sinusoidal form.
Billy Okal
The Kuramoto Model
Introduction The Model Demonstration
Formal Definition
Definition (Kuramoto Model)
The Kuramoto model considers s system of globally coupledoscillators governed by the equation.
dθidt
= ωi +K
NΣNj=1sin(θj − θi ) (1)
where is the phase, ω is the natural frequency, N is the number ofoscillators and K is the coupling constant1
1http:
//www.johnwordsworth.com/tutorials/Kuramoto/index.php?s=4&p=0
Billy Okal
The Kuramoto Model
Introduction The Model Demonstration
Demo
Video and Live Demo
Billy Okal
The Kuramoto Model
