On the number of stable solutions in the Kuramoto model

The Kuramoto model has been extensively studied due to its ability to capture and explain synchronization phenomena in a wide range of systems, including biological, physical, and social systems.^3–6 It has been applied to understand phenomena, such as neuronal synchronization,⁷ power grid synchronization,⁸ and opinion formation in social networks.⁹ 

In this paper, we focus on investigating the stability and properties of equilibrium solutions in the Kuramoto model for a finite system of $n$ oscillators. We analyze the conditions under which stable synchronization emerges and explore the dynamics of the system as the coupling strength $K$ varies. Our findings contribute to a deeper understanding of synchronization mechanisms in finite oscillator networks.

The network’s topology is represented by an adjacency matrix $A$ of size $n×n$⁠. Specifically, $a i i=0$⁠, and $a i j= a j i=1$ if nodes $i$ and $j$ are connected, while $a i j= a j i=0$ otherwise. Furthermore, we require the network to be connected. Systems (3) and (4) described above are known as the “Kuramoto model” of synchronization associated with the network defined by the adjacency matrix $A$⁠. However, we refer to the specific case where $a i j=1$ for all $i≠j$ and $a i i=0$ as the “classic Kuramoto model” since it corresponds to the original formulation.

We observe that the matrix $H(θ)$ is symmetric, which implies that its eigenvalues are real. This property allows us to compute the derivative of the eigenvalues of $H(θ)$ and control how they evolve (as demonstrated in Proposition III.3). Notably, the spectrum of $H(θ)$ remains unaffected by variations in $ω$ or $K$⁠.

To illustrate this claim, let us consider the case where $ω≠ 0$⁠, and without loss of generality, assume that $ω 1≠0$⁠. We observe that the inner product $ω⋅(π,0,…,0)=π ω 1$⁠, whereas $ω⋅(−π,0,…,0)=−π ω 1$⁠. This demonstrates that the inner product does not exhibit $2π$ periodicity, leading to the conclusion that $V$ is not well-defined on the torus $T n$ in the presence of non-zero $ω$⁠.

The Kuramoto model exhibits a finite number of equilibrium points. This can be demonstrated by rewriting the nonlinear system $ω+K f(θ)= 0$ as a quadratic system and applying Bézout’s theorem (for more details, refer to Ref. 10). In previous discussions, we have defined an equilibrium solution of the Kuramoto system (5). Now, we introduce the concept of a stable equilibrium solution, as defined in Ref. 11.

One important remark about the above definition is that the requirement for negative definiteness of $H( θ ∗)$ is not feasible in the Kuramoto model. Due to the nature of the equation driving system (5), stable solutions are not isolated. Specifically, if $θ ∗$ is an equilibrium point, then for any angle $α∈ S 1$⁠, $θ ∗+α=( θ 1 ∗+α,…, θ n ∗+α)$ is also an equilibrium point. We denote all these solutions as $[ θ ∗]= { θ ∗ + α , α ∈ S 1 }$⁠. This fact is evident in the eigenvalues of $H(θ)$⁠, as $1=(1,…,1)$ is an eigenvector of $H(θ)$ with eigenvalue $λ=0$ for all $θ∈ T n$⁠. Thus, $λ=0$ always appears in the spectrum of $H(θ)$⁠. The third condition requires that the remaining eigenvalues of $H( θ ∗)$ are strictly negative.

A second important remark concerns the stability of the equilibrium point $θ ∗$⁠. On one hand, the existence of a strictly positive eigenvalue of $D f( θ ∗)$ implies that the unstable manifold of $θ ∗$ has a dimension greater than or equal to one. On the other hand, the stability of $θ ∗$ can also be linked to the local behavior of $V$ near $θ ∗$⁠. Thus, we impose that $V$ exhibits a local minimum at $θ ∗$⁠. Consequently, the Hessian map $HV( θ ∗)$ is positive semi-definite, and, therefore, $H( θ ∗)$ is negative semi-definite since $HV( θ ∗)=−K H( θ ∗)$ [see Eq. (10)].

The following definition will play a fundamental role in the classification of stable solutions of the Kuramoto model.

The literature on the Kuramoto model is extensive, and it has served as the foundation for the study of various synchronization phenomena. Providing a comprehensive list of all contributions would be impractical. However, we can outline some key findings in the finite Kuramoto model based on different scenarios involving the frequency vector $ω$⁠.

The first case corresponds to $ω= 0$⁠. Taylor¹² demonstrated that the origin $θ ∗=(0,…,0)$ is the only stable solution in the classic Kuramoto model. Subsequently, several authors^12–14 showed that for sufficiently dense networks, the origin is the unique stable solution. Network density is measured using a parameter $μ$⁠, which indicates that each oscillator has at least $μ(n−1)$ connections with other oscillators. The classic Kuramoto model corresponds to $μ=1$⁠. Taylor¹² proved that the origin is the unique stable solution for networks with density parameter $μ≥0.9395$⁠. Ling et al.¹³ established the same result for $μ≥(3− 2)/2∼0.7929$ and Kassabov et al.¹⁴ for $μ≥0.75$⁠. Therefore, it is possible to have multiple stable solutions for small values of $μ$⁠.¹⁵ When $ω= 0$⁠, the Kuramoto model can be analyzed using Morse theory developed by Milnor.¹⁶ In this framework, the system of ODEs (5) represents the downhill flow of the map $V: T n→ R$ (8). However, the global behavior becomes more complex, as Morse theory reveals the existence of multiple unstable solutions. The number of these unstable solutions is related to the Betti numbers of $T n$⁠.¹⁰ For instance, since $V: T n→ R$ is a continuous map defined on a compact manifold, it must reach at least one local maximum. Consequently, there always exist initial conditions that do not converge to the stable solution $θ ∗= 0$⁠. Additionally, it should be noted that in the case of $ω= 0$⁠, the Kuramoto model is independent of parameter $K$⁠.

In the case where $ω≠ 0$⁠, parameter $K$ plays a crucial role in the dynamics of the Kuramoto model (5). Numerical simulations reveal the existence of a critical parameter $K c>0$⁠, such that for $0<K< K c$⁠, the Kuramoto model does not possess any stable solutions. This fact can be easily demonstrated. Let $ω=( ω 1,…, ω n)≠ 0$⁠, and assume without loss of generality that $ω 1≠0$⁠. Then, there exists $ϵ 0$ such that for $0<K< ϵ 0$⁠, we have $ω 1+K f 1(θ)≠0$⁠, since $f 1$ is a bounded map. Therefore, the Kuramoto model (5) does not exhibit any stable solutions for $0<K< ϵ 0$⁠. Numerous results have been obtained regarding the estimation of the critical parameter $K c$ (see Refs. 3 and 17, and references therein).

On the other hand, when parameter $K$ is sufficiently large, the Kuramoto model becomes similar to the case when $ω= 0$⁠. Thus, for large enough $K$⁠, the Kuramoto model possesses a stable equilibrium $θ K$ that converges to $0$ as $K$ tends to infinity. However, the number of stable equilibrium points in the Kuramoto model is still unknown. The stability of solutions in the Kuramoto model has been extensively studied by various authors, including Refs. 11, 18, and 19.

The objective of this work is to investigate the number of stable solutions in the Kuramoto model. Our main result can be stated as follows:

It is important to note that the above result does not impose any explicit assumptions on the system variables, such as the frequencies $ω$⁠, the parameter $K$⁠, or the adjacency matrix $A$⁠.

The Kuramoto model is known to exhibit the possibility of multiple stable solutions (see Ref. 15, and references therein). In Sec. III, we provide a concrete example where the Kuramoto model demonstrates two stable solutions. Furthermore, we discuss how this example relates to the tools utilized in the proof of Theorem A.

The following lemma is well known and can be proven using the Gershgorin circle theorem (see Chap. 6 of Ref. 20). We include it here for completeness.

In our work, the derivative of an eigenvalue with respect to a real parameter will play a fundamental role. The following result can be used to compute the derivative of a simple or multiple eigenvalue with respect to a real parameter. It is a combination of two theorems: Theorem 5 of Ref. 21 for the derivative of a simple eigenvalue and Theorem 2.3 of Ref. 22 for the derivative of a multiple eigenvalue depending on a single real parameter. Additional references related to this result include Refs. 22 and 23.

Using the above result, we can prove that, under certain conditions, the derivative of the eigenvalues of the matrix $H(tθ)$ is not negative. In the following lemma, we collect this result.

It is easy to see that $A(0)$ is a symmetric, diagonally dominant, and semi-definite negative matrix. So, all its eigenvalues are nonpositive (see Lemma III.1). As we travel through the ray $tθ$ for $t∈(0,1/2]$⁠, the spectrum of $A(tθ)$ moves to the right with respect to the spectrum of $A(0)$⁠.

Proof of Theorem A

We start showing that $C$ is an open and nonempty set. We first prove that $0=(0,…,0)$ belongs to $C$⁠. Taking $t=0$ the matrix $H( 0)=A(0)$ (15) is a symmetric, diagonally dominant, and semi-definite negative matrix. Moreover, $λ=0$ is a simple eigenvalue of $H( 0)$ since the network formed by all the oscillators is connected (see Remark 1). We claim that $S$ is an open set. Let $θ 0$ be a point in $C$⁠. We denote by $p θ 0(x)$ the characteristic polynomial of $H( θ 0$⁠). By hypothesis $p θ 0(x)=x⋅ q θ 0(x)$ with $q θ 0(0)≠0$⁠. Moreover, all the roots of $q θ 0(x)$ are real and strictly negative numbers. Hence, in a sufficiently small neighborhood of $θ 0$⁠, the roots of $q θ(x)$ are still strictly negative, showing that $C$ is an open set.

We denote by $∂ C$ and $C ¯$ the boundary and the closure of $C$⁠, respectively. Set $∂ C$ contains all the $θ$’s such that $H(θ)$ is negative semi-definite and $λ=0$ is a multiple eigenvalue. Furthermore, the set $C ¯$ coincides with the set of points where the map $V$ is a convex map. Finally, the open set $[ − π , π ] n∖ C ¯$ contains all the $θ$’s such that $H(θ)$ has at least one strictly positive eigenvalue.

Without loss of generality, we can assume that $C 1$ is the connected component of $C$ containing the origin $0$⁠. A priori we do not know how many of those $C i$ for $i>1$ are different from the empty set. We assume that $θ ∗=( θ 1,…, θ n)$ and $η ∗=( η 1,…, η n)$ are two stable solutions of the Kuramoto model (5) with $|Γ( θ ∗) |≤π$ and $|Γ( η ∗) |≤π$⁠. As we mention before $θ ∗+(α,…,α)$ and $η ∗+(β,…,β)$ are also stable solutions for all $α∈(−π,π]$ and $β∈(−π,π]$⁠. We select $α 0$ and $β 0$ such that $0≤ θ i+ α 0≤π$ and $0≤ η i+ β 0≤π$ for all $1≤i≤n$⁠. So, we just choose a stable solution in the upper half part of the $n$-torus. Thus, renaming $θ ∗=( θ 1,…, θ n)$ and $η ∗=( η 1,…, η n)$⁠, if necessary, we can assume without loss of generality that $θ i, η i∈[0,π]$ for all $1≤i≤n$⁠.

We first assume that $θ ∗$ and $η ∗$ belong to two different connected components. Thus, one of them, for example, $η ∗$ belongs to $C i ∗$ with $i ∗>1$⁠. We consider the ray $t η ∗$ for $t∈[0,1]$⁠. This ray crosses (at least) two different connected components $C 1$ and $C i ∗$⁠. The first one since the origin $0$ is contained in $C 1$ and the second one since $η ∗$ belongs to $C i ∗$⁠. Thus, we can assume that $t η ∗∈ C 1$ for $t∈[0, t 0)$ and $t η ∗∈ C i ∗$ for $t∈( t 1,1]$⁠. Moreover, $t 0 η ∗∈∂ C 1$ and $t 1 η ∗∈∂ C i ∗$⁠. We have proved that the derivative of any eigenvalue $λ(t)$ of $H(t η ∗)$ verifies $λ ′(t)≥0$ (see Proposition III.3).

Now, suppose that the ray $t η ∗$ exits the set $C$ and enters $[ − π , π ] n∖ C ¯$ for some $t∈( t 0, t 1)$⁠. This is a contradiction with the fact that $λ ′(t)≥0$ for $t∈(0,1]$⁠, since at least one eigenvalue needs to be non-negative in $C 1$⁠, then positive in the complement of $C ¯$ and then again non-negative in $C i ∗$⁠. On the other hand, suppose that the ray $t η ∗$ does not exist $C ¯$⁠. This could be the case, for example, if $t 0= t 1$⁠. We observe that in this case when $t→ t 0 −$ all the eigenvalues of $H(t η ∗)$ increase and (at least) one of them collides to $λ=0$ since for $t= t 0$ and for $t→ t 1 −$⁠, this eigenvalue needs to come back, so the derivative at this eigenvalue needs to be strictly negative.

We further assume that $θ ∗$ and $η ∗$ belong to the same connected component. In this scenario, both minima correspond to the same point, as it is implausible to have two distinct local minima in a region where the map is convex, unless $[ θ ∗]=[ η ∗]$⁠.

We conclude this section with two particular examples involving $n=5$ oscillators as related to Theorem A. The first example illustrates a Kuramoto model with a unique stable solution $θ 1$ such that $|Γ( θ 1) |<π$⁠. The second example demonstrates a Kuramoto model with two stable solutions, $θ 1$ and $θ 2$⁠, satisfying $|Γ( θ 1) |<π$ and $|Γ( θ 2) |>π$⁠, respectively.

In this example, we assume that the vector of frequencies $ω$ is zero and $K=1$⁠. The Kuramoto model has two equilibrium points located at $0=(0,0,0,0,0)$ and $θ ∗= ( 0 , 2 π 5 , 4 π 5 , 6 π 5 , 8 π 5 )$⁠. These equilibrium points are stable since the matrices $H( 0)$ and $H( θ ∗)$ have 4 strictly negative eigenvalues, as shown in Ref. 15.

In Fig. 4, we have to plot the graph of the five eigenvalues of $A(t)= H(t θ ∗)$ for $0≤t≤1$⁠. Thus, for $t=0$⁠, the eigenvalues of $H( 0)$ are given by $λ 1(0)=0, λ 2(0)= λ 5(0)=− 1 2 ( 5 + 5 )∼−3.618034$⁠, and $λ 3(0)= λ 4(0)= 1 2 ( 5 − 5 )∼−1.381966$⁠, proving, thus, that $0$ is a stable equilibrium point. Similarly, for $t=1$⁠, the eigenvalues of $H( θ ∗)$ are given by $λ 1(1)=0$⁠, $λ 2(1)= λ 5(1)=− 1 2 ( 5 + 5 )cos⁡ ( 2 π 5 )∼−1.118034$ and $λ 3(1)= λ 4(1)= 1 2 ( 5 − 5 )cos⁡ ( 2 π 5 )∼−0.427051$ showing that $θ ∗$ is also a stable equilibrium solution of the Kuramoto model.