Begin by considering a system of N globally coupled differential equations with stable limits cycles [6]. Note that we are only going to consider the normal form calculation in this derivation as this suffices for our purpose. We could define this system of equations as follows;
The functions fi(xi) are ε close and define the natural dynamics of this dimension in the system. We then add a coupling parameter, with coupling strength ε/N where we choose ε and divide this by the number of oscillators in the system. Finally, g is a Phase Response Curve defining the interaction of the system. Under the assumption that |ε| << 1 we can rewrite the above equation as;
Where Γ() is the convolution of the PRC with the interaction terms evaluated along the limit cycle. The Kuramoto model assumes that this system has a sinusoidal PRC [6] so next consider a specific example giving us this.
The functions fi(xi) are ε close and define the natural dynamics of this dimension in the system. We then add a coupling parameter, with coupling strength ε/N where we choose ε and divide this by the number of oscillators in the system. Finally, g is a Phase Response Curve defining the interaction of the system. Under the assumption that |ε| << 1 we can rewrite the above equation as;
Where Γ() is the convolution of the PRC with the interaction terms evaluated along the limit cycle. The Kuramoto model assumes that this system has a sinusoidal PRC [6] so next consider a specific example giving us this.
