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# Kuramoto

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The Kuramoto Model

The Kuramoto Model

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• 1. The Kuramoto Model Billy Okal April 12, 2012
• 2. Introduction The Model DemonstrationOutline 1 Introduction Oscillators Synchronization 2 The Model 3 DemonstrationBilly OkalThe Kuramoto Model
• 3. Introduction The Model DemonstrationOscillatorsOscillators 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 OkalThe Kuramoto Model
• 4. Introduction The Model DemonstrationSynchronizationSynchronization 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 OkalThe Kuramoto Model
• 5. Introduction The Model DemonstrationFormal 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 OkalThe Kuramoto Model
• 6. Introduction The Model DemonstrationFormal 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=0Billy OkalThe Kuramoto Model
• 7. Introduction The Model DemonstrationDemo Video and Live DemoBilly OkalThe Kuramoto Model