Ordering Parameter

Definition

Ordering parameter is a scalar quantity which is often used to describe the degree of synchronization that a population of oscillators achieve. To better understand this conception, here we consider the most popular version of the Kuramoto model, where each of the oscillators is considered to have its own intrinsic natural frequency ${\omega_i}$, and each is coupled equally to all other oscillators:

$$ \begin{array}{ll}\frac{{d{\theta_i}}}{{dt}} & = {\omega_i} + \frac{K}{N}\sum\limits_{{j = 1}}^N {\sin \left( {{\theta_j} - {\theta_i}} \right)}, { }i \\ & = 1,2, \cdots, N \end{array} $$

(1)

where the system is composed of N limit-cycle oscillators.

Noise can be added to the system. In that case, the original equation is altered to

$$ \begin{array}{ll}\frac{{d{\theta_i}}}{{dt}} & = {\omega_i} + {\xi_i} + \frac{K}{N}\sum\limits_{{j = 1}}^N {\sin \left( {{\theta_j} - {\theta_i}} \right)}, { }i \\ & = 1,2, \cdots, N \end{array}$$

(2)

where ${\xi_i}$is the fluctuation...