Stability of Clusters in the Second-Order Kuramoto Model on Random Graphs

Abstract

The Kuramoto model of coupled phase oscillators with inertia on Erdős–Rényi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

Appendix A: \(K_c\) is an Increasing Function of \(\sigma \)

Suppose g zero mean Gaussian density with standard deviation $\sigma$

$$\begin{aligned} g(s) = \frac{1}{\sigma \sqrt{2\pi }}{\text {exp}}\left( -\frac{s^2}{2\sigma ^2}\right) . \end{aligned}$$

(A.1)

By plugging (A.1) into (4.5) we have

$$\begin{aligned} K_c(\sigma ) = 2\sqrt{2\pi }\sigma \left( \pi -\displaystyle \int _{\mathbb {R}} \frac{{\mathrm{exp}}\left( -\frac{s^2}{2\sigma ^2}\right) }{\gamma ^2+(s/\gamma )^2}ds\right) ^{-1}, \end{aligned}$$

which can be further rewritten as

$$\begin{aligned} K_c(\sigma ) = \frac{2\sqrt{2}\sigma }{\sqrt{\pi }}\left( 1-{\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right) \right) ^{-1}, \end{aligned}$$

(A.2)

where ${\text {Erfc}}(x) = 1- \frac{2}{\sqrt{\pi }}\int _0^x e^{-s^2}ds$ is the function.

Lemma A.1

$K_c$ in (A.2) is an increasing function of $\sigma >0$.

Proof

Let $A = {\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right)$ and note that

$$\begin{aligned} \frac{dK_c}{d\sigma } = \frac{2\sqrt{2}}{\sqrt{\pi }}\left( (1-A)+\sigma (1-A)^{-2}\frac{dA}{d\sigma }\right) . \end{aligned}$$

Below we show that $A<1$ and $\frac{dA}{d\sigma }>0$ for all $\sigma >0$. Assume first that $\sigma < \gamma ^2$. Then, using the well known bound

$$\begin{aligned} {\text {Erfc}}(z) < \frac{{\text {exp}}(-z^2)}{\sqrt{\pi }z}, \end{aligned}$$

(A.3)

it follows that $A< \frac{\sqrt{2}{\sigma }}{\sqrt{\pi }\gamma ^2} <1$.

Next consider when $\sigma \ge \gamma ^2.$ Since the Taylor series expansion of ${\text {Erfc}}(z)$ is an alternating series

$$\begin{aligned} {\text {Erfc}}(z) = 1-\frac{2}{\sqrt{\pi }} \left( z-\frac{z^3}{3}+\frac{z^5}{10}-\cdots \right) , \end{aligned}$$

(A.4)

we have

$$\begin{aligned} {\text {Erfc}}(z) < 1 -\frac{2}{\sqrt{\pi }} z + \frac{2}{3\sqrt{\pi }} z^3. \end{aligned}$$

(A.5)

Thus,

$$\begin{aligned} A < {\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) \left( 1-\frac{\sqrt{2}}{\sqrt{\pi }}\cdot \frac{\gamma ^2}{\sigma } + \frac{1}{3\sqrt{2\pi }} \cdot \frac{\gamma ^6}{\sigma ^3}\right) . \end{aligned}$$

(A.6)

In this case we have

$$\begin{aligned} A< e^{1/2}\left( 1-\frac{\sqrt{2}}{\sqrt{\pi }}+\frac{1}{3\sqrt{2\pi }}\right) <1. \end{aligned}$$

(A.7)

Finally, we show that $\frac{dA}{d\sigma }>0$. Indeed, by direct calculation,

$$\begin{aligned} \frac{dA}{d\sigma } = \frac{\sqrt{2}\gamma ^2}{\sqrt{\pi }\sigma ^2 }- \frac{\gamma ^4}{\sigma ^3}{\text {exp}}\left( \frac{\gamma ^4}{2\sigma ^2}\right) {\text {Erfc}}\left( \frac{\gamma ^2}{\sigma \sqrt{2}}\right) . \end{aligned}$$

Again use (A.3) to see that

$$\begin{aligned} \frac{dA}{d\sigma } > \frac{\sqrt{2}\gamma ^2}{\sqrt{\pi }\sigma ^2}-\frac{\gamma ^4}{\sigma ^3}\cdot \frac{\sqrt{2}\sigma }{\sqrt{\pi }\gamma ^2} =0 . \end{aligned}$$

(A.8)

$\square$