We denote each part \(i\) by its position on a circle (which corresponds to the angle \(\theta_i\), from \(0^{\circ}\) to \(360^{\circ}\), which rotates with its own frequency \(\omega_i\). Without any coupling, the angle at time \(t\) can be calculated as \(\theta_i(t) = \omega_i t + \theta_i(0)\). The coupling between each part is described by the sinus of the angle difference. The matrix \(A\) describes which oscillator is connected to each other. In the classical Kuramoto model, this matrix is identical to one everywhere.

The full systems equation is given by:

\[\begin{aligned} \frac{d\theta_i}{d t} = \omega_i - \frac{K}{N} \sum_j \sin A_{ij} \left( \theta_i - \theta_j\right)\end{aligned}\]

Where \( \frac{d\theta_i}{d t}\) describes change in time of the variable \(\theta_i\). For identical angles \(\theta_i\), the sinus term is zero and the system evolve in synchronized manner, similar as described before.

As observed before, the number of equilibrium points changes depending on the coupling strength. This qualitative change in behavior is called a bifurcation. Bifurcations are an important behavior in many dynamical systems and have been studied extensively in chaos theory, as they are closely linked to the emergence of chaotic behavior. Bifurcations are often studied by their bifurcation diagram, which shows the equilibrium points with respect to a variable. The bifurcation diagram with respect to the coupling strength looks like this:

The observed bifurcation is also called a Saddle-node bifurcation.

Here, we formalize the argumentation about equilibrium points and stability of the previous chapter, for the interested reader:

We describe a dynamical system by its state variables \(x\) (in general, a vector). An important feature of dynamical systems is that their behavior only depend on the current state. Continuous dynamical systems are usually described by a set of (ordinary) differential equations, given as: \[\begin{aligned} \frac{d}{dt} x = f(x)\end{aligned}\]

Often, their behavior is studied by the equilibrium points \(x^*\), which are the solution of: \[\begin{aligned} f(x^*) = 0\end{aligned}\]

We investigate the stability by investigating the behavior of a vector \(x = x^* + \Delta x\), with a small pertubation \(\Delta x\). We assume that \(\Delta x\) is positive, the same argument can be made for negative \(\Delta x\) (with inverted signs).

We can then study the behavior by Taylor-expanding \(f\) and get: \[\begin{aligned} f(x) \approx \underbrace{f(x^*)}_{=0} + \sum_j \underbrace{\frac{\partial f(x^*)}{\partial x_j}}_{=J} \underbrace{\left( x_j - x_j^* \right)}_{=\Delta x_j} \\ \text{With the Jacobian matrix: } J = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} &\dots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1} &\dots & \frac{\partial f_n}{\partial x_n} \end{pmatrix}\end{aligned}\]

We can then calculate the stability by the eigenvalues of the Jacobian matrix at the fixed point (\(J(x^*)\)). Negative eigenvalues correspond to stable fixed points.

We try to calculate a critical value for the existence of a synchronized solution of the Kuramoto model. As before, the phases of the Kuramoto model is given as:

\[\begin{aligned} \frac{d\theta_i}{d t} = \omega_i - \frac{K}{N} \sum_j \sin \left( \theta_i - \theta_j\right)\end{aligned}\]

We define the order parameter \(r\) and the average phase \(\psi\) with the imaginary unit \(i = \sqrt{-1}\) as:

\[\begin{aligned} r e^{i\psi} = \frac{1}{N} \sum_j e^{i \theta_j}\end{aligned}\]

If we multiply this formula by \(e^{-i \theta_i}\), we can express the Kuramoto model as:

\[\begin{aligned} \frac{d\theta_i}{d t} = \omega_i - K r \sin \left( \theta_i - \psi\right)\end{aligned}\]

It should be noted that \(0 \leq r \leq 1\). The synchronized solution is found for \(r = 1\), when all phases are identical.

We can express a necessary condition for a critical coupling coefficient by looking into the synchronized state \(\frac{d\theta_i}{d t} = \frac{d\theta_j}{d t}\):

\[\begin{aligned} 0 = \frac{d\theta_i}{d t} - \frac{d\theta_j}{d t} = \omega_i - \omega_j - K r \underbrace{\left( \sin(\left(\theta_i - \psi\right) - \sin(\left(\theta_j - \psi\right) \right)}_{\leq 2}\end{aligned}\]

Which has to be true for all phases \(\theta_i\). If we consider that the order parameter \(r \leq 1\), we cannot find any any solutions for the maximum case if \(K < \left(\omega_i - \omega_j\right)/2\). Thus, a value for the critical coupling \(K_{\text{crit}}\) is given as:

\[\begin{aligned} K_{\text{crit}} \geq \frac{\left( \omega_i - \omega_j \right)}{2}\end{aligned}\]

This is only a necessary (thus not sufficiently or tight!) condition for coupling. If we consider all oscillators, then we arrive at \(K_{\text{crit}} \geq \frac{\left( \omega_{\text{max}} - \omega_{\text{min}} \right)}{2}\)

The properties of a network can be studied by the matrix representations of the graph. One common matrix representation is the Laplacian matrix \(L\). The Laplacian matrix is defined by using the edge weight of nodes \(i,j\) at the corresponding indices and the (weighted) nodal degrees at the main diagonal. From the definition, it follows that the Laplacian matrix is a singular matrix. For a undirected graph, it is a symmetrical matrix (and thus has real eigenvalues). The eigenvalues are actually all positive. If the Graph is connected, only one eigenvalue is zero. The second smallest eigenvalue is the so called algebraic connectivity \(\lambda_2\). The algebraic connectivity is a measure of the connective-ness of a network. The algebraic connectivies of the graphs used in the applets are shown in the following table:

Can you observe how the algebraic connectivity relates to synchronization?