We consider a system of coupled oscillators described by the Kuramoto model with the dynamics given by . In this system, an equilibrium solution is considered stable when , and the Jacobian matrix has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that , where represents the length of the shortest arc on the unit circle that contains the equilibrium solution . We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system.
Synchronization in ensembles of network-coupled heterogeneous oscillators is crucial in various natural and engineered phenomena, ranging from cell cycles to robust power systems. One of the most prominent and elegant models for studying synchronization is the Kuramoto model,1 which provides a mathematically tractable description of this phenomenon. Kuramoto recognized the mean-field approach as the most suitable method for analytical treatment and introduced an all-to-all purely sinusoidal coupling scheme, deriving the governing equations for each oscillator in the system. The Kuramoto model is a mathematically tractable description of synchronization in network-coupled heterogeneous oscillators. We investigate the conditions for stable equilibrium solutions in this model. Our main finding is that a stable equilibrium solution exists when , and the Jacobian matrix has zero as a simple eigenvalue and negative eigenvalues in orthogonal directions. We also establish the constraint , indicating that the equilibrium lies within the shortest arc on the unit circle containing it. This analysis contributes to understanding the dynamics and stability of 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 oscillators. We analyze the conditions under which stable synchronization emerges and explore the dynamics of the system as the coupling strength varies. Our findings contribute to a deeper understanding of synchronization mechanisms in finite oscillator networks.
II. PRELIMINARIES AND MAIN RESULT
The network’s topology is represented by an adjacency matrix of size . Specifically, , and if nodes and are connected, while 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 . However, we refer to the specific case where for all and as the “classic Kuramoto model” since it corresponds to the original formulation.
We observe that the matrix is symmetric, which implies that its eigenvalues are real. This property allows us to compute the derivative of the eigenvalues of and control how they evolve (as demonstrated in Proposition III.3). Notably, the spectrum of remains unaffected by variations in or .
To illustrate this claim, let us consider the case where , and without loss of generality, assume that . We observe that the inner product , whereas . This demonstrates that the inner product does not exhibit periodicity, leading to the conclusion that is not well-defined on the torus in the presence of non-zero .
The Kuramoto model exhibits a finite number of equilibrium points. This can be demonstrated by rewriting the nonlinear system 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.
We say that is a stable solution of (5) if and only if
is negative semi-definite, and
One important remark about the above definition is that the requirement for negative definiteness of is not feasible in the Kuramoto model. Due to the nature of the equation driving system (5), stable solutions are not isolated. Specifically, if is an equilibrium point, then for any angle , is also an equilibrium point. We denote all these solutions as . This fact is evident in the eigenvalues of , as is an eigenvector of with eigenvalue for all . Thus, always appears in the spectrum of . The third condition requires that the remaining eigenvalues of are strictly negative.
A second important remark concerns the stability of the equilibrium point . On one hand, the existence of a strictly positive eigenvalue of implies that the unstable manifold of has a dimension greater than or equal to one. On the other hand, the stability of can also be linked to the local behavior of near . Thus, we impose that exhibits a local minimum at . Consequently, the Hessian map is positive semi-definite, and, therefore, is negative semi-definite since [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 .
The first case corresponds to . Taylor12 demonstrated that the origin is the only stable solution in the classic Kuramoto model. Subsequently, several authors12–14 showed that for sufficiently dense networks, the origin is the unique stable solution. Network density is measured using a parameter , which indicates that each oscillator has at least connections with other oscillators. The classic Kuramoto model corresponds to . Taylor12 proved that the origin is the unique stable solution for networks with density parameter . Ling et al.13 established the same result for and Kassabov et al.14 for . Therefore, it is possible to have multiple stable solutions for small values of .15 When , 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 (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 .10 For instance, since 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 . Additionally, it should be noted that in the case of , the Kuramoto model is independent of parameter .
In the case where , parameter plays a crucial role in the dynamics of the Kuramoto model (5). Numerical simulations reveal the existence of a critical parameter , such that for , the Kuramoto model does not possess any stable solutions. This fact can be easily demonstrated. Let , and assume without loss of generality that . Then, there exists such that for , we have , since is a bounded map. Therefore, the Kuramoto model (5) does not exhibit any stable solutions for . Numerous results have been obtained regarding the estimation of the critical parameter (see Refs. 3 and 17, and references therein).
On the other hand, when parameter is sufficiently large, the Kuramoto model becomes similar to the case when . Thus, for large enough , the Kuramoto model possesses a stable equilibrium that converges to as 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:
Let and be two stable solutions of the Kuramoto model (5) satisfying and . Then, .
It is important to note that the above result does not impose any explicit assumptions on the system variables, such as the frequencies , the parameter , or the adjacency matrix .
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.
III. PROOF OF THEOREM A AND EXAMPLES
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.
Let be an real, symmetric and diagonal dominant matrix, i.e., for . The following conditions hold:
Assume that all the entries in the diagonal verify . Then, all the eigenvalues of are non-negative, and, therefore, is positive semi-definite and for all vectors .
Assume that all the entries in the diagonal verify . Then, all eigenvalues of are non-positive, and, therefore, is negative semi-definite and for all vectors .
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 is not negative. In the following lemma, we collect this result.
Let be any point in (2). For any , we define the matrix . Let be an eigenvalue of . Then, for all . Moreover, assuming that for all , then for all .
We claim that is a symmetric, diagonally dominant, and positive semi-definite matrix for all . The symmetry of follows from the evenness of the function and the symmetry of the adjacency matrix .
By hypothesis, is a point in the -torus , where for all . Consequently, for all . Therefore, for any , we have for all . It follows that for any
Now, let us consider the case where satisfies for all . From this assumption, we can easily conclude that for all . As a result, for all where is the auxiliary function defined earlier (see Fig. 2). Consequently, matrix is symmetric, diagonally dominant, and positive semi-definite for all .
Finally, let be any eigenvalue of . By applying Lemma III.2, we have , since is a positive semi-definite matrix for any .
It is easy to see that 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 for , the spectrum of moves to the right with respect to the spectrum of .
The Laplacian matrix is a matrix representation of a network. In particular, the rank of the Laplacian matrix is related to the number of connected components of the network (see Chap. 13 of Ref. 24 for details). Let be the Laplacian matrix associated to the network of oscillators. From the expression of (15), we just observe that . Moreover, it is well known that is an eigenvalue of whose multiplicity coincides with the number of connected components of the graph (Lemma 13.1.1 of Ref. 24). In our case, we have assumed that our network of oscillators form a connected graph. So, we conclude that is a simple eigenvalue of and the rest of its eigenvalues are strictly negative real numbers.
Proof of Theorem A
We start showing that is an open and nonempty set. We first prove that belongs to . Taking the matrix (15) is a symmetric, diagonally dominant, and semi-definite negative matrix. Moreover, is a simple eigenvalue of since the network formed by all the oscillators is connected (see Remark 1). We claim that is an open set. Let be a point in . We denote by the characteristic polynomial of ). By hypothesis with . Moreover, all the roots of are real and strictly negative numbers. Hence, in a sufficiently small neighborhood of , the roots of are still strictly negative, showing that is an open set.
We denote by and the boundary and the closure of , respectively. Set contains all the ’s such that is negative semi-definite and is a multiple eigenvalue. Furthermore, the set coincides with the set of points where the map is a convex map. Finally, the open set contains all the ’s such that has at least one strictly positive eigenvalue.
Without loss of generality, we can assume that is the connected component of containing the origin . A priori we do not know how many of those for are different from the empty set. We assume that and are two stable solutions of the Kuramoto model (5) with and . As we mention before and are also stable solutions for all and . We select and such that and for all . So, we just choose a stable solution in the upper half part of the -torus. Thus, renaming and , if necessary, we can assume without loss of generality that for all .
We first assume that and belong to two different connected components. Thus, one of them, for example, belongs to with . We consider the ray for . This ray crosses (at least) two different connected components and . The first one since the origin is contained in and the second one since belongs to . Thus, we can assume that for and for . Moreover, and . We have proved that the derivative of any eigenvalue of verifies (see Proposition III.3).
Now, suppose that the ray exits the set and enters for some . This is a contradiction with the fact that for , since at least one eigenvalue needs to be non-negative in , then positive in the complement of and then again non-negative in . On the other hand, suppose that the ray does not exist . This could be the case, for example, if . We observe that in this case when all the eigenvalues of increase and (at least) one of them collides to since for and for , this eigenvalue needs to come back, so the derivative at this eigenvalue needs to be strictly negative.
We further assume that and 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 .
We conclude this section with two particular examples involving oscillators as related to Theorem A. The first example illustrates a Kuramoto model with a unique stable solution such that . The second example demonstrates a Kuramoto model with two stable solutions, and , satisfying and , respectively.
In this example, we assume that the vector of frequencies is zero and . The Kuramoto model has two equilibrium points located at and . These equilibrium points are stable since the matrices and have 4 strictly negative eigenvalues, as shown in Ref. 15.
In Fig. 4, we have to plot the graph of the five eigenvalues of for . Thus, for , the eigenvalues of are given by , and , proving, thus, that is a stable equilibrium point. Similarly, for , the eigenvalues of are given by , and showing that is also a stable equilibrium solution of the Kuramoto model.
In Proposition III.3, we have proved that all the eigenvalues verify for . In Fig. 4, we also have to plot the vertical line and we can see that the functions are increasing for all as it is proved in Proposition III.3. Moreover, it is possible to prove that and for some value of .
Although Theorem A does not apply to this particular example (since ), the ideas used in the proof of Theorem A can used to understand the existence of these two stable equilibrium solutions. Thus, these two stable equilibrium solutions and belong to two different connected components of set [see (16)]. We recall that is the set of points where four of the eigenvalues of are strictly negative. We denote by the connected component of containing and the connected component of containing . In Fig. 4, it is shown the evolution of all the eigenvalues from to . We observe that is the only eigenvalue that changes their sign. More precisely, for and for (see Fig. 4). Hence, at , point belongs to the boundary of , and for , point is the complement of and at , point belongs to the boundary of and for , point belongs to . Thus, eigenvalue needs to increase to exit and decrease to enter into or in other words their derivative change their sign.
In this paper, we have demonstrated that the Kuramoto model yields a unique stable solution satisfying . This implies that half of the unit circle can be selected to include all the oscillators. Such solutions can be perceived as a type of “entrained” solution, characterizing what is commonly seen as a cluster of entrained oscillators around a mutual phase. This is distinct from other solution types, such as the splay state solution depicted in Fig. 3. The stability of any equilibrium solution of (5) is captured within the symmetric matrix as detailed in Eq. (6). Our proof of this principal finding hinges on controlling the derivative of the eigenvalues of the function , where denotes an eigenvalue of the matrix .
A.A. and S.G. also acknowledge support from Spanish Ministerio de Ciencia e Innovacion (No. PID2021-128005NB-C21), Generalitat de Catalunya (Nos. 2021SGR-633 and PDAD14/20/00001), and Universitat Rovira i Virgili (No. 2019PFR-URV-B2-41). A.G. and J.V. also acknowledge support from the Ministry of Science, Innovation and Universities of Spain through Grant No. MTM 2020-118281GB-C33. A.A. also acknowledges support from ICREA Academia, and the James S. McDonnell Foundation (220020325), the Joint Appointment Program at Pacific Northwest National Laboratory (PNNL). PNNL is a multi-program national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-AC05-76RL01830, and the European Union’s Horizon Europe Programme under the CREXDATA project, Grant Agreement No. 101092749.
Conflict of Interest
The authors have no conflicts to disclose.
Alex Arenas: Conceptualization (lead); Formal analysis (equal); Supervision (lead); Writing – review & editing (equal). Antonio Garijo: Formal analysis (lead); Supervision (equal); Writing – original draft (lead). Sergio Gómez: Formal analysis (equal); Writing – review & editing (equal). Jordi Villadelprat: Formal analysis (equal); Supervision (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.