Explosive Synchronization and Multistability in Large Systems of Kuramoto Oscillators with Higher-Order Interactions

Abstract

Since its original formulation, the Kuramoto model and its many variants have served as critical tools for uncovering and understanding the emergence of nonlinear collective behavior. However, recent evidence suggests that in such phase-reduced systems, interactions beyond the typical pair-wise angle differences need to be considered to develop a full picture of the dynamics. In particular, higher-order interactions, namely non-additive, nonlinear interactions that take place between three or more oscillators are required. Here we explore these interactions and their effect on the macroscopic dynamics of coupled phase oscillator systems. The analysis for these systems begins with all-to-all coupled systems where a range of techniques including dimensionality reductions and self-consistency analyses may be employed. The effects of the various higher-order coupling terms on the macroscopic dynamics may then be explored, revealing a natural mechanism for nonlinear phenomena that includes abrupt (i.e., explosive) synchronization transitions and extensive multistability. These dynamics are qualitatively preserved under more heterogeneous network topologies. Moreover, the high degree of multistability in such networks allows for the system to store information and memory.

8.1 Introduction

The Kuramoto model of coupled phase oscillators has guided researchers interested in studying collective behavior in nonlinear dynamical systems for decades [1]. The classical formulation involves N all-to-all coupled phase oscillators with states $\theta _n$, $n=1\dots ,N$, whose dynamics are governed by

$$\begin{aligned} \dot{\theta }_i=\omega _i+\frac{K}{N}\sum _{j=1}^N\sin (\theta _j-\theta _i), \end{aligned}$$

(8.1)

where $\omega _n$ is the natural frequency of oscillator n, K is the global coupling strength, and the sine of the respective phase angles represents the effect of oscillator m on oscillator n. The degree of synchronization of the system is typically measured using Kuramoto’s order parameter,

$$\begin{aligned} z=re^{i\psi }=\frac{1}{N}\sum _{j=1}^Ne^{i\theta _m}, \end{aligned}$$

(8.2)

which gives a complex number that represents the centroid of all phases when placed appropriately on the complex unit circle. In particular, the magnitude $r=|z|$ takes values in between zero and one with $r\approx 0$ and $r\approx 1$ representing incoherent and synchronized states, respectively. In cases where multiple clusters emerge a collection of higher-order variants of the order parameters, also known as the Daido order parameters, are used to capture the macroscopic dynamics and are given by

$$\begin{aligned} z_q=r_qe^{i\psi _q}=\frac{1}{N}\sum _{j=1}^Ne^{qi\theta _m}, \end{aligned}$$

(8.3)

where q is a natural number and $q=1$ recovers the classical Kuramoto order parameter given by Eq. (8.2). As we will see in Sect. 8.3, the collection of order parameters $z_1,\dots ,z_m$ are useful for characterizing the synchronization in a system with m clusters. In addition to the all-to-all coupled system described given in Eq. (8.1) the dynamics of the network-coupled analogue, given by

$$\begin{aligned} \dot{\theta }_i=\omega _i+K\sum _{j=1}^NA_{ij}\sin (\theta _j-\theta _i), \end{aligned}$$

(8.4)

has been used to explore the many effects of network structure on collective behavior.

It is important to note that the original system given by Eq. (8.1) is the result of a phase-reduction from a system of globally-coupled limit cycle oscillators, where the state of a limit-cycle oscillator is assumed to be well-approximated by a single phase angle. Interestingly, recent work that aims to study these phase reductions beyond first-order contributions have identified higher-order interactions, i.e., non-additive, nonlinear interactions between three or more oscillators, as important components for describing the full system dynamics [2, 3]. These theoretical studies are complemented by empirical evidence in neuroscience that suggest higher-order interactions may play an important role in brain function [4,5,6]. In particular, three such higher-order interactions emerge and as an analogue to Kuramoto sine-coupling in Eq. (8.1) take the following forms:

$$\begin{aligned} \text {Type I}:&~~\sum _{j=1}^N\sum _{l=1}^N\sin (2\theta _j-\theta _l-\theta _i), \end{aligned}$$

(8.5)

$$\begin{aligned} \text {Type II}:&~~\sum _{j=1}^N\sum _{l=1}^N\sin (\theta _j+\theta _l-2\theta _i), \end{aligned}$$

(8.6)

$$\begin{aligned} \text {Type III}:&~~\sum _{j=1}^N\sum _{l=1}^N\sum _{m=1}^N\sin (\theta _j +\theta _l-\theta _m-\theta _i), \end{aligned}$$

(8.7)

where type I and II coupling naturally take place between sets of three oscillators (i.e., triangles or 2-simplexes) and type III coupling takes place between sets of four oscillators (i.e., tetrahedra or 3-simplexes). In the remainder of this chapter we explore the dynamics that emerge in phase oscillator systems as a result of incorporating these higher-order interaction terms, both together with the classical pairwise coupling term and on their own. In Sect. 8.2 we study a system that includes pairwise coupling along with type I and III coupling and show that the higher-order coupling terms give rise to nonlinear behavior that allows for abrupt synchronization transitions, hysteresis, and stable synchronized states even for negative pairwise-coupling. In Sect. 8.3 we study type II coupling in isolation and show that it leads to abrupt desynchronization transitions (without a complementary mechanism for synchronization) and an extensive multistability that allows the system to store memory. In Sect. 8.4 we provide some initial investigation into the effects of nontrivial network structures. Lastly, we briefly provide an outlook for future work in Sect. 8.5.

8.2 Coupling Types I and III: Abrupt Synchronization

When incorporating type I and III interactions into the classical Kuramoto model we obtain the following higher-order Kuramoto model originally studied in Ref. [7]:

$$\begin{aligned} \dot{\theta }_i=\omega _i&+\frac{K_1}{N}\sum _{j=1}^N\sin (\theta _j-\theta _i)+\frac{K_2}{N^2}\sum _{j=1}^N\sum _{l=1}^N\sin (2\theta _j-\theta _l-\theta _i)\nonumber \\ {}&+ \frac{K_3}{N^3}\sum _{j=1}^N\sum _{l=1}^N\sum _{m=1}^N\sin (\theta _j+\theta _l-\theta _m-\theta _i), \end{aligned}$$

(8.8)

where coupling strengths $K_1$, $K_2$, and $K_3$ measure the strength of interactions between pairs, triangles, and tetrahedra of oscillators (i.e., 1-simplexes, 2-simplexes, and 3-simplexes, respectively). The choice of coupling terms used in Eq. (8.8) conveniently allows us to solve exactly for the macroscopic system dynamics in the continuum limit $N\rightarrow \infty$ using the Ott-Antonsen ansatz [8,9,10]. In particular, we note that by defining

$$\begin{aligned} H=K_1z+K_2z_2z^*+K_3z^2z^* \end{aligned}$$

(8.9)

where $*$ denotes the complex conjugate (and recall that $z=z_1$), we may rewrite Eq. (8.8) as

$$\begin{aligned} \dot{\theta }_i=\omega _i+\frac{1}{2i}\left( He^{-i\theta _i}-H^*e^{i\theta _i}\right) . \end{aligned}$$

(8.10)

Next, in the continuum limit we may describe the state of the system using the density function $f(\theta ,\omega ,t)$, where $f(\theta ,\omega ,t)d\theta d\omega$ gives the fraction of oscillators with phase in $[\theta ,\theta +d\theta )$ and natural frequency in $[\omega ,\omega +d\omega )$ at time t. The conservation of oscillators and the static natural frequencies implies that f satisfies the continuity equation $0=\frac{\partial f}{\partial t}+\frac{\partial }{\partial \theta }\left( f\dot{\theta }\right)$, i.e.,

$$\begin{aligned} 0=\frac{\partial f}{\partial t}+\frac{\partial }{\partial \theta }\left\{ f\left[ \omega _i+\frac{1}{2i}\left( He^{-i\theta _i}-H^*e^{i\theta _i}\right) \right] \right\} . \end{aligned}$$

(8.11)

Since $\theta$-domain of f is the circle it is natural to consider the Fourier expansion of f as

$$\begin{aligned} f(\theta ,\omega ,t)=\frac{g(\omega )}{2\pi }\left[ 1+\sum _{n=1}^\infty \hat{f}_n(\omega ,t)e^{in\theta }+\sum _{n=1}^\infty \hat{f}_n^*(\omega ,t)e^{-in\theta }\right] . \end{aligned}$$

(8.12)

Remarkably, Ott and Antonsen discovered that the choice of geometrically-decaying Fourier coefficients, i.e., $\hat{f}_n(\omega ,t)=\alpha ^n(\omega ,t)$ where $\alpha$ is assumed to be an analytic function in the $\omega$-complex plane, drastically reduces the system dynamics. In particular, inserting this choice into Eq. (8.12) and then into Eq. (8.11) yields a single ordinary differential equation for $\alpha$, in this case given by

$$\begin{aligned} \dot{\alpha }=-i\omega \alpha +\frac{1}{2}\left( H^*-H\alpha ^2\right) . \end{aligned}$$

(8.13)

It is worth noting that this family of solutions with geometrically-decaying Fourier coefficients yields density functions $f(\theta ,\omega ,t)$ that are Poison kernels and describes a stable manifold to which all solutions tend [9, 10]. To close the system dynamics we note that the order parameter satisfies

$$\begin{aligned} z^*=\iint f(\theta ,\omega ,t)e^{i\theta }d\theta d\omega =\int \alpha (\omega ,t)g(\omega )d\omega . \end{aligned}$$

(8.14)

A convenient choice for the frequency distribution g is the Lorentzian with mean $\omega _0$ and width $\Delta$, i.e., $g(\omega )=\Delta /{\pi [\Delta ^2+(\omega -\omega _0)^2]}$. The integral in Eq. (8.14) can then be evaluated by closing the contour with the infinite-radius semi-circle in the negative-half complex plane and using Cauchy’s integral theorem, yielding $z^*=\alpha (\omega _0-i\Delta ,t)$. Similarly, for the order parameter $z_2$ we have that $z_2^*=\alpha ^2(\omega _0-i\Delta )=z^{*2}$, and by evaluating Eq. (8.13) at $\omega =\omega _0-i\Delta$ and taking a complex conjugate we find

$$\begin{aligned} \dot{z}=-\Delta z+i\omega _0 z+\frac{1}{2}\left[ \left( K_1z+K_{2+3}z^2z^*\right) -\left( K_1z^*+K_{2+3}z^{*2}z\right) z^2\right] , \end{aligned}$$

(8.15)

where we have defined the combined higher-order coupling strength $K_{2+3}=K_2+K_3$. Finally, it is convenient to decompose the complex dynamics given in Eq. (8.15) into amplitude and angle components, yielding

$$\begin{aligned} \dot{r}&= -\Delta r + \frac{K_1}{2}r(1-r^2)+\frac{K_{2+3}}{2}r^3(1-r^2), \end{aligned}$$

(8.16)

$$\begin{aligned} \dot{\psi }&=\omega _0. \end{aligned}$$

(8.17)

Equations (8.16) and (8.17) close the dynamics for the macroscopic system dynamics and allow us to now perform a bifurcation and stability analysis of the synchronization dynamics of the higher-order Kuramoto model in the continuum limit. Unsurprisingly, the amplitude and angle dynamics decoupling, leaving states to rotate with a constant frequency equal to the mean natural frequency of the system. Inspecting Eq. (8.16) more closely, we also observe that the contribution of the higher-order interaction terms only contribute higher-order nonlinear terms implying that they have no effect on the stability of the incoherent state $r=0$. In fact, $r=0$ is always a steady-state solution to Eq. (8.16) and is stable for $K_1<2\Delta$, and loses stability at the critical value $K_1=2\Delta$ in a pitchfork bifurcation that may be supercritical or subcritical, depending on the amplification of the nonlinear terms by $K_{2+3}$. The nature of this bifurcation is most easily seen by solving for synchronized steady-states, namely, $r>0$ and $\dot{r}=0$, yielding

$$\begin{aligned} r=\sqrt{\frac{K_{2+3}-K_1\pm \sqrt{(K_1+K_{2+3})^2-8\Delta K_{2+3}}}{2K_{2+3}}}. \end{aligned}$$

(8.18)

The transition between incoherence and synchronization can be identified by investigating how this synchronized branch meets the incoherent branch $r=0$ in particular, as r tends towards zero $K_1$ tends towards $2\Delta$ regardless of $K_{2+3}$, but for $K_{2+3}\le 2\Delta$ and $K_{2+3}>2\Delta$ it tends towards $2\Delta$ from above and below, respectively, indicating a supercritical to subcritical shift. This can be seen most easily in Fig. 8.1a where we plot the synchronized branch as a function of $K_1$ for $K_{2+3}$ values 0, 2, 5, 8, and 10 (blue ranging to red) and $\Delta = 1$. The curves represent the exact prediction given in Eq. (8.18) while circles correspond to simulations of $N=10^4$ oscillators. In particular, the synchronized branch begins to fold over itself at $K_{2+3}=2\Delta$, indicating supercritical and subcritical pitchfork bifurcations for $K_{2+3}\le 2\Delta$ and $K_{2+3}>2\Delta$, respectively. A straight forward analysis reveals that, when the synchronized branch folds over itself the plus and minus signs in Eq. (8.18) give stable and unstable solutions, respectively, which are depicted in solid and dashed curves, respectively. To complement Fig. 8.1a we plot in Fig. 8.1b the synchronized branch as a function of $K_{2+3}$ for $K_{1}$ values $-0.5$, 1, 1.8, 2, and 2.2 (blue ranging to red) and $\Delta = 1$.

Fig. 8.1

(Modified and adapted from Ref. [7])

Abrupt synchronization in the higher-order Kuramoto model. Synchronization profiles describing the macroscopic system state: the order parameter r a as a function of 1-simplex coupling $K_1$ for higher-order coupling $K_{2+3}=0$, 2, 5, 8, and 10 (blue to red) and b as a function of higher-order coupling $K_{2+3}$ for 1-simplex coupling $K_1=-0.5$, 1, 1.8, 2, and 2.2. Solid and dashed curves represent stable and unstable solutions given by Eq. (8.18), respectively, and circles denote results taken from direct simulations of equation (8.8) with $N=10^4$ oscillators with $\Delta = 1$ and $\omega _0=0$.

Fig. 8.2

(Modified and adapted from Ref. [7]

Stability diagram for the higher-order Kuramoto model. The full stability diagram describing incoherent, synchronized, and bistable states as a function of 1-simplex coupling $K_1$ and higher-order coupling $K_{2+3}$ for $\Delta = 1$. Blue and red curves correspond to pitchfork and saddle-node bifurcations, which collide at a codimension-two point (black circle) at $(K_1,K_{2+3})=(2,2)$. For $K_{2+3}<2$ and $K_{2+3}>2$ the pitchfork bifurcation is supercritical and subcritical, respectively.

This reveals, for $K_{2+3}>2\Delta$, a region of bistability between the incoherent and synchronized states in the form of a hysteresis loop bounded by two critical values. The first of these critical values is the synchronization value $K_1^{\text {sync}}=2\Delta$ where the incoherent state loses stability and the system undergoes an abrupt transition to a synchronized state. The second critical value occurs at a smaller value of $K_1$ located at the left-most point of the synchronized branch, at which point, as $K_1$ is decreased, the synchronized branch annihilates in a saddle-node bifurcation and the system undergoes an abrupt transition to the incoherent state. Solving for this second desynchronization value, we obtain $K_1^{\text {desync}}=2\sqrt{2\Delta K_{2+3}}-K_{2+3}$, which collides with $K_1^{\text {sync}}=2\Delta$ at $K_{2+3}=2\Delta$, corresponding to the codimension-two point where the pitchfork bifurcation transitions between subcritical and supercritical. Setting $\Delta = 1$ (which can be done without loss of generality by rescaling time and the parameters $K_1$, $K_{2+3}$ and $\omega _0$) we plot in Fig. 8.2 the stability diagram for the higher-order Kuramoto model indicating incoherent, synchronized, and bistable regions between pitchfork and saddle-node curves (blue and red, respectively) and the codiension-two point at $(K_1,K_{2+3})=(2,2)$. Interestingly, as the saddle-node curve crosses into the negative-half plane $K_1<0$ the system is able to support synchronized states even for negative pairwise coupling, as was observed in Fig. 8.1a.

8.3 Coupling Type II: Abrupt Desynchronization and Multistability

Next we investigate the dynamics in the presence of only type II interactions, yielding the following model originally studied in Ref. [11]:

$$\begin{aligned} \dot{\theta }_i = \omega _i + \frac{K}{N^2}\sum _{j=1}^N\sum _{l=1}^N\sin \left( \theta _{j}+\theta _{l}-2\theta _i\right) . \end{aligned}$$

(8.19)

As we will see in what follows, the collective dynamics under type II interactions deserve special attention for two reasons. First, the arrangement of terms in the coupling function (i.e., compared to that in type I coupling) makes complicates the analysis in that the Ott-Antonsen dimensionality reduction does not fully solve the system–instead they provide a partial dimensionality reduction, beyond which a self-consistency approach is required. Second, the effects of type II coupling on the collective dynamics differ significantly from those explored above under type I and III interactions. One such effect is the emergence of cluster synchronization, where oscillators become entrained in two different clusters on the torus centered at opposite angles. We note here that by entering a suitable rotating frame we may set the mean natural frequency to zero, ensuring non-rotating entrained solutions, and moreover a shift in initial conditions ensures that the largest cluster (i.e., that with the largest fraction of entrained oscillators) is centered at $\theta =0$ with the smaller cluster centered opposite at $\theta =\pi$.

Fig. 8.3

(Modified and adapted from Ref. [11])

Collective dynamics under type II coupling. The order parameters a $r_1$ and b $r_2$ as a function of the coupling strength K for various asymmetry values $\eta$. Blue, red, green, orange, and purple circles represent results obtained from direct simulations of Eq. (8.19) with $N=10^5$ oscillators with natural frequencies drawn from a Lorentzian with $\omega _0=0$ and $\Delta = 1$ for $\eta =1$, 0.95, 0.9, 0.85, and 0.8, respectively

The formation of these two clusters and the particular form of coupling then lead to unexpected nonlinear behaviors which we illustrate now with numerical simulations of $N=10^5$ oscillators with natural frequencies drawn from a Lorentzian distribution with mean $\omega _0=0$ and width $\Delta = 1$. We then introduce an asymmetry parameter $\eta$ that describes the initial conditions: at time $t=0$ we start with a randomly chosen fraction of $\eta$ of all oscillators at phase $\theta =0$, while a fraction $(1-\eta )$ of the oscillators start at $\theta =\pi$. We then simulate the system dynamics by adiabatically decreasing the coupling strength from $K=16$ to 0, then adiabatically increase it again until we reach $K=16$ again. The results are shown in Fig. 8.3, where the values of $r_1$ and $r_2$ are plotted in panels (a) and (b), respectively, as a function of the coupling strength K, for asymmetry parameters $\eta = 1$ (blue), 0.95 (red), 0.9 (green), 0.85 (orange), and 0.8 (purple). As seen in both plots the system begins in a synchronized state and, as K is decreased, the degree of synchronization decreases continuously until, at a critical value of K that depends on the asymmetry $\eta$, the system undergoes an abrupt desynchronization transition. The state then remains incoherent until K reaches zero. Then, as K is increased and restored to 16 the system surprisingly remains incoherent, i.e., there is no synchronization transition (abrupt or continuous) that complements the abrupt desynchronization transition. Moreover, further numerical simulations (not shown) and the analysis presented below confirm that no such synchronization transitions exist above $K=16$. In addition to these abrupt desynchronization transitions, the simulations above show that, for a given coupling strength, many stable entrained states are possible, depending on the asymmetry $\eta$. In fact, in the thermodynamic limit of $N\rightarrow \infty$ a continuum of entrained states may be stable, indicating an extensive multistability.

We now present an analysis to explain these dynamics. First, we use the definition of the order parameter to rewrite Eq. (8.19) as

$$\begin{aligned} \dot{\theta }_i=\omega _i + K r_1^2\sin [2(\psi _1-\theta _i)]. \end{aligned}$$

(8.20)

As is the previous section, we consider the continuum limit $N\rightarrow \infty$ where the state of the system can be described by a density function $f(\theta ,\omega ,t)$ that satisfies the same continuity equation $0=\partial _t f + \partial _\theta (f\dot{\theta })$. In contrast to the case with type I and III interactions, here we must consider the symmetric and asymmetric parts $f_s(\theta ,\omega ,t)$ and $f_a(\theta ,\omega ,t)$, respectively, of $f(\theta ,\omega ,t)$ which satisfy

$$\begin{aligned} f(\theta ,\omega ,t)=f_s(\theta ,\omega ,t)+f_a(\theta ,\omega ,t), \end{aligned}$$

(8.21)

with symmetries

$$\begin{aligned} f_s(\theta +\pi ,\omega ,t)=f_s(\theta ,\omega ,t),~~~\text {and}~~~f_a(\theta +\pi ,\omega ,t)=-f_a(\theta ,\omega ,t). \end{aligned}$$

(8.22)

Note that the linearity of the continuity equation implies that if both $f_s$ and $f_a$ are solutions, then so is f. While the asymmetric part $f_a$ may not be simplified by dimensionality reduction, we may simplify the symmetric part $f_s$. Noting that the Fourier series of $f_s$ is given by the even terms of the Fourier series of f, i.e.,

$$\begin{aligned} f_s(\theta ,\omega ,t)=\frac{g(\omega )}{2\pi }\left[ 1+\sum _{m=1}^\infty \hat{f}_{2m}(\omega ,t)e^{2im\theta }+c.c.\right] , \end{aligned}$$

(8.23)

we propose an ansatz similar to that above where these even Fourier coefficients decay geometrically, i.e., $\hat{f}_{2m}(\omega ,t)=a^m(\omega ,t)$. Inserting this and Eq. (8.23) and then into the continuity equation, we find that the symmetric dynamics collapse onto the same low-dimensional manifold characterized by the condition

$$\begin{aligned} \partial _t a=-2i\omega a+K\left( z_{1}^{*2}-z_{1}^2 a^2\right) . \end{aligned}$$

(8.24)

As was done above, the symmetric dynamics may be closed by expressing $a(\omega ,t)$ as an order parameter as follows. In particular, assuming again that that the frequency distribution $g(\omega )$ is Lorentzian, $g(\omega )=\Delta /\{\pi \left[ (\omega -\omega _0)^2+\Delta ^2\right] \}$ a similar technique as that used in the previous section below Eq. (8.15) results in $z_2=a^*(\omega _0-i\Delta ,t)$. We then evaluate Eq. (8.24) at $\omega =\omega _0-i\Delta$ to obtain

$$\begin{aligned} \dot{z}_2=2i\omega _0z_2-2\Delta z_2+K\left( z_{1}^2-z_{1}^{*2}z_2^2\right) , \end{aligned}$$

(8.25)

or, in polar form,

$$\begin{aligned} \dot{r}_2&=-2\Delta r_2+Kr_{1}^2(1-r_2^2)\cos (2\psi _{1}-\psi _2), \end{aligned}$$

(8.26)

$$\begin{aligned} \dot{\psi }_2&=2\omega _0+Kr_{1}^2\frac{1+r_2^2}{r_2}\sin (2\psi _{1}-\psi _2). \end{aligned}$$

(8.27)

Equations (8.26) and (8.27) describe the symmetric dynamics captured by the order parameter $z_2$. However, these equations do not capture the asymmetric part of the dynamics, and moreover they depends on the asymmetric dynamics via $z_2$’s dependence on $z_1$. To complete the analysis we next need a self consistency analysis. Recalling our assumptions where $\omega _0=0$ and clusters set at $\theta =0$ and $\pi$, effectively setting the order parameter phases at $\psi _1,\psi _2=0$, we revisit Equation (8.20) which implies that oscillators that become phase-locked satisfy $|\omega _i|\le Kr_1^2$, in which case they relax to one of the two stable fixed points $\theta _i=\theta ^*(\omega _i)$ or $\theta ^*(\omega _i)+\pi$, where

$$\begin{aligned} \theta ^*(\omega )=\arcsin (\omega /Kr_1^2)/2. \end{aligned}$$

(8.28)

These two fixed points correspond to the two clusters to which that the phase-locked oscillators become entrained – specifically, phase-locked oscillators starting near $\theta =0$ or $\pi$ will end up at the fixed points $\theta ^*(\omega )$ or $\theta ^*(\omega )+\pi$, respectively. The phase-locked population is described by the density function

$$\begin{aligned} f_{\text {locked}}(\theta ,\omega )=\eta \delta (\theta -\theta ^*(\omega )) + (1-\eta )\delta (\theta -\theta ^*(\omega )-\pi ), \end{aligned}$$

(8.29)

where the asymmetry parameter $\eta$ appears and describes the fraction of phase-locked oscillators in the $\theta =0$ cluster. On the other hand, oscillators satisfying $|\omega _i|>Kr_1^2$ drift for all time and relax to the following stationary distribution

$$\begin{aligned} f_{\text {drift}}(\theta ,\omega ) = \frac{\sqrt{\omega ^2-K^2r_1^4}}{2\pi \left[ \omega +Kr_1^2\sin (2\psi _1-2\theta )\right] }. \end{aligned}$$

(8.30)

Next, the order parameter $z_1$ is given by the integral $z_1=\iint f(\theta ,\omega ,t)e^{i\theta }d\theta d\omega$, which after inserting the density f as defined by Eqs. (8.29) and (8.30) reduces to

$$\begin{aligned} r_1=(2\eta -1)\int \limits _{-Kr_1^2}^{Kr_1^2}\sqrt{\frac{1+\sqrt{1-(\omega /Kr_1^2)^2}}{2}}g(\omega )d\omega , \end{aligned}$$

(8.31)

where the contribution from the drifting oscillators vanishes due to the symmetry of $f_{\text {drift}}$. Returning to $r_2$, Eq. (8.26) implies that at steady state we have

$$\begin{aligned} r_2 = \frac{-\Delta + \sqrt{\Delta + K^2r_1^4}}{Kr_1^2}. \end{aligned}$$

(8.32)

Thus, the macroscopic steady-state is described by Eqs. (8.31) and (8.32).

Fig. 8.4

(Modified and adapted from Ref. [11])

Synchronized states with type II coupling. For order parameters a $r_1$ and b $r_2$, the synchronization branches predicted by Eqs. (8.31) and (8.32) plotted as solid curves and results from simulations plotted in circles. Results correspond to asymmetries $\eta =1$, 0.95, 0.9, 0.85, and 0.8, as well as the minimum branches [predictions for which are given in Eq. (8.35)]. Simulated results use $N=10^5$ oscillators

In Fig. 8.4 we overlay the analytical predictions of Eqs. (8.31) and (8.32) (solid curves) with the synchronized states observed in simulations (circles) for the same asymmetry parameter values $\eta$ used in Fig. 8.3, noting excellent agreement. [Equation (8.31) was solved numerically to find $r_1$.] We also investigate the smallest $r_1$ and $r_2$ supported by the system by, for each K, decreasing $\eta$ until the synchronized state is lost. These simulation results are plotted in black circles. Interestingly, while these $r_1^{\text {min}}$ values decrease as K increases, $r_2^{\text {min}}$ appears to remain roughly constant. In fact, we now use this surprising property to uncover the critical desynchronization coupling strength as a function of the asymmetry parameter, i.e., $K_c(\eta )$, or equivalently, the minimum asymmetry parameter that supports a synchronized state as a function of coupling, i.e., $\eta _{\text {min}}(K)$. We begin by first inverting Eq. (8.32), obtaining $Kr_1^2=2\Delta r_2/(1-r_2^2)$, which can be inserted into Eq. (8.31), yielding

$$\begin{aligned}&\sqrt{\frac{2\Delta r_2}{1-r_2^2}}=\sqrt{K}(2\eta -1)\int \limits _{-2\Delta r_2/(1-r_2^2)}^{2\Delta r_2/(1-r_2^2)}\sqrt{\frac{1+\sqrt{1-\left[ \omega (1-r_2^2)/2\Delta r_2\right] ^2}}{2}}g(\omega )d\omega . \end{aligned}$$

(8.33)

While Eq. (8.11) appears more complicated than Eq. (8.9), we note that the coupling strength K has been entirely scaled out of the integral, appearing outside with $(2\eta -1)$. We therefore conclude that if the quantities $\sqrt{K}$ and $2\eta -1$ cancel one another out, i.e., $\sqrt{K}(2\eta -1)$ is constant, it follows that the solution $r_2$ in Eq. (8.11) is independent of K. We therefore ansatz that $\sqrt{K}(2\eta -1)=\text {const.}$ and use the initial condition $\eta _{\text {min}}(K_c(1))=1$, where $K_c(1)$ denotes the very first coupling strength where a synchronized state is possible with $\eta =1$, yielding

$$\begin{aligned} \eta _{\text {min}}(K)=\frac{\sqrt{K_c(1)}}{2\sqrt{K}}+\frac{1}{2}. \end{aligned}$$

(8.34)

Equation (8.34) implies that along the minimum branch we have that $\sqrt{K}(2\eta -1)=\sqrt{K_c(1)}\approx 2.034$, which can be used in Eq. (8.32) to compute the minimum branch of $r_2$, and in turn $r_1$ via Eq. (8.31), yielding

$$\begin{aligned} r_1^{\text {min}}(K)\approx \frac{1.2120}{\sqrt{K}}\text {and}~r_2^{\text {min}}(K)\approx 0.5290. \end{aligned}$$

(8.35)

These predictions are plotted as black curves in Fig. 8.4. Lastly, by inverting Eq. (8.34) we find the critical coupling strength $K_c$ as a function of $\eta$ where the abrupt desynchronization transition occurs, namely

$$\begin{aligned} K_c(\eta ) \approx \frac{4.137}{(2\eta -1)^2}. \end{aligned}$$

(8.36)

In Fig. 8.5 we plot the theoretical prediction of the abrupt desynchronization point $K_c(\eta )$ as a solid curve vs observations from direct simulations as black circles, noting excellent agreement. Above this curve we observe extensive multistability and below the curve only the incoherent state is stable. Some additional rigorous results that agree and complement these may be found in Refs. [12, 13].

Fig. 8.5

(Modified and adapted from Ref. [11])

Abrupt desynchronization transition. The critical coupling strength $K_c$ at which the abrupt desynchronization transition occurs as a function of the asymmetry $\eta$. The solid curve represents the theoretical prediction given by Eq. (8.36) and black circles represent observations from direct simulation with $N=10^5$ oscillators

8.4 Effects of Network Structure

Having presented results on synchronization in the presence of higher-order interactions in globally-coupled systems, we now turn our attention to the effects on network structure. Here we focus on the network analogue of the higher-order Kuramoto model presented in Sect. 8.2. In the presence of a non-trivial network topology the system dynamics are described by

$$\begin{aligned} \dot{\theta }_i=\omega _i&+\frac{K_1}{\langle k^1\rangle }\sum _{j=1}^NA_{ij}\sin (\theta _j-\theta _i)+\frac{K_2}{2\langle k^2\rangle }\sum _{j=1}^N\sum _{l=1}^NB_{ijl}\sin (2\theta _j-\theta _l-\theta _i)\nonumber \\&+ \frac{K_3}{6\langle k^3\rangle }\sum _{j=1}^N\sum _{l=1}^N\sum _{m=1}^NC_{ijlm}\sin (\theta _j+\theta _l-\theta _m-\theta _i), \end{aligned}$$

(8.37)

where all dynamical variables and parameters are the same as in Sect. 8.2 except for those that are network-dependent. In particular, he network structure (assumed to be undirected and unweighted) is encoded in the 1-simplex adjacency matrix A, 2-simplex adjacency tensor B, and 3-simplex adjacency tensor C, where $A_{ij}=1$ if nodes i and j are connected by a link (and otherwise $A_{ij}=0$), $B_{ijl}=1$ if nodes i, j, and l belong to a common 2-simplex (and otherwise $B_{ijl}=0$), and $C_{ijlm}=1$ if nodes i, j, l, and m belong to a common 3-simplex (and otherwise $C_{ijlm}=0$). For each node i we denote the q-simplex degree $k_i^q$ as the number of distinct q-simplexes node i is a part of, and $\langle k^q\rangle$ is the mean q-simplex degree across the network. Importantly, the division of each coupling strength by the respective mean degree $\langle k^q\rangle$ in Eq. (8.37) simply amounts to a rescaling of the respective coupling strength, and moreover ensures that the mean-field approximation of the dynamics of Eq. (8.37) are given precisely by Eq. (8.8).

Fig. 8.6

(Modified and adapted from Ref. [7])

Higher-order Kuramoto model: Real networks. The order parameter r as a function of 1-simplex coupling $K_1$ for the network analogue of the higher-order Kuramoto model for a the Macaque brain network and b the UK power grid. In each case $K_1$ is adiabatically increased then decreased to highlight the bistable region. Higher-order coupling is given by a $K_{2}=1.6$ and $K_3=1.1$ and b $K_{2}=2.2$ and $K_3=3.3$

Fig. 8.7

(Modified and adapted from Ref. [7])

Higher-order Kuramoto model: Multiplex model. The order parameter r a as a function of 1-simplex coupling $K_1$ for higher-order coupling $K_{2+3}=0$, 2, 5, 8, and 10 (blue to red) and b as a function of higher-order coupling $K_{2+3}$ for 1-simplex coupling $K_1=-0.5$, 1, 1.8, 2, and 2.2. Circles represent direct simulations on a network of $N=10^4$ nodes with mean degrees $\langle k^1\rangle =\langle k^2\rangle =\langle k^3\rangle =30$ and solid and dashed curves represent stable and unstable solutions of the mean field approximation.

We begin with direct simulations of Eq. (8.37) on two real-world networks: the Macaque brain network [14] and the UK power grid [15]. In the Macaque brain network we infer higher-order connection directly as triplets (i, j, l) and quadruplets (i, j, l, m) that are fully-connected cliques. Using natural frequencies drawn from the standard normal distribution, we set higher-order coupling strengths $K_{2}=1.6$ and $K_3=1.1$ and adiabatically increase and decrease $K_1$ and plot the order parameter r as a function of $K_1$ in Fig. 8.6a. For the UK power grid we follow a similar protocol with $K_{2}=2.2$ and $K_3=3.3$, plotting the results in Fig. 8.6b, but given the geographical nature of the network, we incorporate a triplet or quadruplet interaction for any connected three- or four-path in the network. Importantly, in these real-world topologies we see qualitatively similar dynamics as in the globally coupled system presented in Sect. 8.2. We note, however, that while the results are qualitatively similar, more quantitative predictions of the dynamics remain inaccurate.

Next we turn our attention to a generative network model, in particular, a multiplex model where interactions of different order are uncorrelated. Specifically, setting target mean degrees $\langle k^1\rangle$, $\langle k^2\rangle$, and $\langle k^3\rangle$, we place at random $M_1=N\langle k^1\rangle /2$ 1-simplexes (i.e., links), $M_2=N\langle k^1\rangle /3$ 2-simplexes (i.e., filled triangles), and $M_3=N\langle k^3\rangle /4$ 3-simplexes (i.e., filled tetrahedra). Note that in this model the interactions of different orders are uncorrelated, so that, for example, a pair of nodes may be part of a triangle even if there exists no link between them and vice versa. Thus, this model may be thought of as a three-layer multiplex with 1-, 2-, and 3-simplexes belonging to respective layers. In Fig. 8.7 we plot the results of the dynamics of such a multiplex model of $N=10^4$ oscillators with mean degrees $\langle k^1\rangle =\langle k^2\rangle =\langle k^3\rangle =30$, plotting r as a function of $K_1$ for $K_{2+3}=0$, 2, 5, 8, and 10 in panel (a) and as a function of $K_{2+3}$ for $K_1=-0.5$, 1, 1.8, 2, and 2.2 in panel (b). simulation results are plotted in circles with solid curves representing the prediction from the mean-field approximation, i.e., Eq. (8.18), showing strong agreement.

8.5 Outlook and Future Work

We now close with a brief outlook to future work. As we have seen in this chapter, the presence of higher-order interactions in systems of coupled oscillators introduces many new dynamics effects in synchronization dynamics. Even in the globally-coupled case there are many avenues to explore that promise rich dynamics. Examples include (but are certainly not limited to) the incorporation of time delays, phase lags, positive/negative coupling, external forcing—all of which have profound effects on the collective dynamics in the absence of higher-order interactions. For the higher-order Kuramoto model studied in Sect. 8.2 the viability of the Ott-Antonsen ansatz promises to be very useful, but the technical peculiarities of type II coupling complicate the analysis of this future work.

Further progress in the case of nontrivial network topologies is also needed. One such case would be the presence of modular structure in networks and how it affects the dynamics explored above. More generally, however, the generic effects of non-trivial network structures are poorly understood. In Sect. 8.4 we saw that while the real networks displayed qualitatively similar dynamics as the globally-coupled system, quantitative predictions were poor. On the other hand, the dynamics of the multiplex model were well-captured by the mean field analysis, suggesting that correlations between interactions of different orders plays in important role. Another broad question lies in the effect of network (i.e., degree) heterogeneity and its affect on collective dynamics, most importantly the abrupt transitions.

Lastly, the multistability induced by type II coupling studied in Sect. 8.3 deserves special mention. This phenomenon allows an oscillator system to store information and memory by treating each oscillator as a bit: 0 or 1 if the oscillator is in the $\theta = 0$ or $\pi$ cluster, respectively. Relatively small perturbations and parameter modifications then allow the system to represent a different string, i.e., piece of information. In the presence of a network topology these effects remain, although the supported (i.e., stable) states changes. Preliminary work [16] shows that sparser networks support fewer states, but the overall effect of network structure remains poorly understood.