### The idea of synchrony of phase oscillators

An oscillator model is a dynamical system in which the state variables evolve through a periodic trajectory or orbit in state space (also called phase space). This periodic orbit is a loop through state space, as the state of the system returns to where it started after some time $T$, called the period of the oscillator.

#### The phase oscillator

A phase oscillator is an oscillator model where we view the state of the system as going around the simplest loop, a unit circle. In a phase oscillator model, the state variable $\theta$ is called the phase of the oscillator. The state space in which the phase $\theta$ lives is the interval $[0, 2\pi]$. To turn the interval into a circle, we make it periodic so that $\theta=0$ is the same state as $\theta=2\pi$. We can view this interval as a unit circle where $\theta$ marks the angle going around the circle.

A simple way to describe the unit circle is by embedding in the complex plane. A complex number $z$ is on the unit circle if it has length one, $|z|=1$. If $|z|=1$, then we can write it as $$z = e^{i\theta},$$ where $i=\sqrt{-1}$ and $\theta$ is the counterclockwise angle between the positive real axis and the vector from the origin to the complex number $z$, as demonstrated in the below applet.

*The phase of an oscillator.* We can represent the state of an oscillator by its *phase* $\theta(t)$, which lies in the interval $[0,2\pi]$. Since we view the phase as being a periodic variable, we can draw the interval $[0,2\pi]$ as a circle in the complex plane where $\theta$ is the angle between the positive real axis (what you might call the positive $x$-axis) and the vector from the origin to a point on the unit circle. This point is drawn in green, and you can change it by dragging it with the mouse. To display the phase most compactly, we will typically draw the phase of an oscillator as just a point on the unit circle (and omit the vector and angle shown here). The coordinates of the point in the complex plane are given by $e^{i\theta(t)}$ where $i=\sqrt{-1}$.

#### Synchrony of phase oscillators

If we have a collection of phase oscillators, one measure of the collective behavior of the oscillators is their degree of synchrony. If all the oscillators are going around the circle exactly together, then we say the oscillators are completely synchronized. There are also lesser degrees of synchrony, as the oscillators could also go around the circle more-or-less together but spread out a bit in phase. In contrast, the oscillators could be completely asynchronous, meaning their phases are completely spread around the unit circle.

##### The Kuramoto order parameters

We can use a measure of synchrony to capture the level of synchrony of a collection of phase oscillators. One measure of synchrony is the Kuramoto order parameter. Actually, there are two Kuramoto order parameters, but we will be primarily concerned with the measure $r$ that indicates the level of synchrony. The second order parameter $\psi$ gives the average phase of the collection of oscillators.

We define the order parameters simply by averaging the complex numbers that represent the phase of the oscillators on the unit circle. Given a collection of $N$ phase oscillators with phases $\theta_j$ for $j=1,2, \ldots N$, the positions of the oscillators on the unit circle are represented by the complex numbers $e^{i\theta_j}$. Define the complex number $z$ as the average of these positions $$z = \frac{1}{N}\sum_{j=1}^Ne^{i\theta_j}.$$

To understand what this sum means, you can view each oscillator as being represented by a vector from the origin to the corresponding point on the unit circle (e.g., the red vector in the previous applet). Recalling the geometric definition of vector addition, we can deduce that the sum $z$ can be represented by a vector that points in the average direction of all the $\theta_j$ vectors. If all the phases $\theta_j$ are equal, then their average will be the unit vector pointing to that phase. On the other hand, if two phases are opposite of each other on the unit circle, their average will be the zero vector. Hence the closer the phases are to each other, the longer $z$ will be, and it points in the average direction of the phases. You can experiment with the dependence of $z$ on the phases below.

*The Kuramoto order parameters.* Given a collection of phase oscillators, the Kuramoto order parameters $r$ and $\psi$ capture the degree of their synchrony and the average phase, respectively. The phases $\theta_j$ of $N$ oscillators are shown as green points around the unit circle in the complex plane. You can change $N$ by dragging the red point on the slider and change the phases by dragging the green points. The location of the point corresponding to $\theta_j$ is $e^{i\theta_j}$ where $i=\sqrt{-1}$, so $\theta_j$ measures the counterclockwise angle from the positive real axis (i.e., what you'd think of as the positive $x$-axis). The vector $z$ is the average of the $N$ points viewed as vectors in the complex plane: $z = \frac{1}{N}\sum_{j=1}^N e^{i\theta_j}.$ The order parameter $r$ is the magnitude of $z$, $r=|z|$. The average phase $\psi$ is the argument of $z$, i.e., counterclockwise angle between $z$ and the positive real axis. One can write $z=re^{i\psi}$.

The Kuramoto order parameters are derived from the average vector $z$. The synchrony measure $r$ is the length of $z$, $r=|z|$. It achieves its maximum $r=1$ when all the phases are identical. It achieves its minimum $r=0$ when the phases are balanced around the circle, such as evenly spread or in clusters that balance each other out. The synchrony measure $r$ is often called the *phase coherence*.

Another useful piece of information is the average phase of the oscillators, which is the direction that $z$ points. The second order parameter $\psi$ captures this direction, as it is the counterclockwise angle between $z$ and the positive real axis, also called the argument of $z$. We can represent the complex number $z$ in terms of its length $r$ and argument $\psi$ as $$z = r e^{i\psi}.$$

##### Synchronizing oscillators

To view how the Kuramoto order parameter $r$ captures synchrony, we look at two examples. In this first example, the oscillators begin with phases spread randomly and uniformly over the phase space (the unit circle). As time evolves, the oscillators begin to synchronize. The vector $z$ captures this synchronization by getting larger, with $r$ approaching 1 (unless you degrade the final synchrony by making $\sigma$ larger).

*Synchronizing oscillators.* lllustration of $N=100$ phase oscillators whose state can be represented by a point around the unit circle. The state of oscillator $j$ at time $t$ is represented by the state variable $\theta_j(t)$, or phase, which we can map to the unit circle by plotting a green point in the complex plane at the location $e^{i\theta_j(t)}$ where $i=\sqrt{-1}$. For each oscillator, $\theta_j(t)$ increases with time, so each point moves counterclockwise around the circle. At time $t=0$, the oscillators are randomly spread over state space. However, as time evolves, the oscillators tend to group together to have similar phase, i.e., they tend to synchronize. You can control how well they synchronize by changing the parameter $\sigma$, which controls the final spread of points. When $\sigma=0$, the oscillators would eventually completely synchronize so that all $\theta_j(t)$ become identical. The degree of synchrony is captured by the complex number $z=\frac{1}{N}\sum_{i=1}^N e^{i\theta_j(t)},$ plotted as a blue vector and by its magnitude $r=|z|$, which ranges from 0 to 1.

##### Desynchronizing oscillators

In the second example, we look at the opposite situation. We start the phase oscillators completely synchronized. But, since they oscillate at different frequencies, their phases soon spread out around the phase space. The rate of this desynchronization depends on the the spread of the frequencies (controlled by the parameter $\sigma$). The complex number $z$ and its magnitude $r$ shrink as this desynchronization progresses.

*Desynchronizing oscillators.* A group of $N=100$ phase oscillators represented just as in the previous applet, only this group of oscillators tends to desynchronize. At time $t=0$, all oscillators start at exactly the same phase. However, as time evolves, the oscillators tend to spread out across the whole circle of phases, i.e., the system tends toward asynchrony. You can control how they desynchronize by changing the parameter $\sigma$, which controls the spread of oscillator speeds. When $\sigma=0$, the oscillators will stay synchronized forever.