Phase transition and scaling in Kuramoto model with high-order coupling

Abstract

The classical Kuramoto model serves as a useful tool for studying synchronization transitions in coupled oscillators that is limited to the sinusoidal and pairwise interactions. In this paper, we extend the classical Kuramoto model to incorporate the high-order structures and non-pairwise interactions into the coupling function. Using a self-consistent approach and constructing parametric functions, we describe the extensive multi-cluster states induced by high-order structures and identify various types of phase transitions toward synchrony. In particular, we establish the universal scaling relation for each branch of multiclusters, which describes the asymptotic dependence of the order parameters (Kuramoto and Daido) on the coupling strength near the critical points.

Appendix

In this Appendix, we deduce the scaling behaviors of order parameters close to $K_{c}$ dominated by the linear high-order coupling. Following the similar method, Taylor expansion of H(q) at $q_{c}$ yields

$$\begin{aligned} H(q)=H(q_{c})+A(\delta q)^{\mu }+B(\delta q)^{\nu }, \end{aligned}$$

(A1)

with (A, B) being the expanding coefficients and $(\mu ,\nu )$ being the power indices. Substituting Eq. (A1) into the self-consistent equation Eq. (22), we have

$$\begin{aligned} \delta K=-\frac{K_{c}^{2}A\delta q^{\mu }+K_{c}^{2}B\delta q^{\nu }}{1+K_{c}A\delta q^{\mu }+K_{c}B\delta q^{\nu }}. \end{aligned}$$

(A2)

Similarly, we expand Eq. (A2) as the following Taylor series

$$\begin{aligned} \delta K=X(\delta q)^{a}+Y(\delta q)^{b}. \end{aligned}$$

(A3)

To reversely solve $\delta q$, we adopt the following ansatz

$$\begin{aligned} \delta q=X^{-\frac{1}{a}}\delta K^{\frac{1}{a}}+E\delta K^{\varepsilon }. \end{aligned}$$

(A4)

The undetermined quantities E and $\varepsilon$ can be solved self-consistently, which are the same as Eq. (29).

Since $R_{2}=\frac{q}{K}$ and $R_{1}=(2\eta -1)h(q)$we have

$$\begin{aligned} \delta R_{2}=\frac{\delta q}{K_{c}}-\frac{q_{c}}{K_{c}^{2}}\delta K, \end{aligned}$$

(A5)

and

$$\begin{aligned} \delta R_{1}=(2\eta -1)h^{\prime }(q_{c})\delta q. \end{aligned}$$

(A6)

Substituting Eq. (A4) into Eq. (A5), and if $q_{c}=0$ we have the parameters array

$$\begin{aligned} \chi _{2}=\left( \frac{1}{a},\varepsilon ,\frac{X^{-\frac{1}{a}}}{K_{c}},\frac{E}{K_{c}}\right) . \end{aligned}$$

(A7)

However if $q_{c}>0$, $\chi _{2}$ should be discussed in three different cases according to the value of $\varepsilon$. For $\varepsilon <1$, $\chi _{2}$ is the same as Eq. (A7). If $\varepsilon =1$,

$$\begin{aligned} \chi _{2}=(\frac{1}{a},1,\frac{X^{-\frac{1}{a}}}{K_{c}},(\frac{E}{K_{c}}-\frac{q_{c}}{K_{c}^{2}})), \end{aligned}$$

(A8)

and if $\varepsilon >1$,

$$\begin{aligned} \chi _{2}=(\frac{1}{a},1,\frac{X^{-\frac{1}{a}}}{K_{c}},-\frac{q_{c}}{K_{c}^{2}}). \end{aligned}$$

(A9)

Finally, the parameters array $\chi _{1}$ becomes

$$\begin{aligned} \chi _{1}= & {} \left( \frac{1}{a},\varepsilon ,(2\eta -1)f^{\prime }(q_{c})X^{-\frac{1}{a}},\right. \nonumber \\&\left. (2\eta -1)f^{\prime }(q_{c})E\right) . \end{aligned}$$

(A10)