On the stability of the Kuramoto model of coupled nonlinear oscillators

By Ali Jadbabaie, Nader Motee, Mauricio Barahona. Proceedings of the 2004 American Control Conference, 2005

510 citations recorded by the source corpus.

We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We also provide a series of bounds for the critical values of the coupling strength.

Subfields: Computer Networks and Communications, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/e31e1fa572d34419757ddc5bf7be54c437edbc5f

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Citations in the corpus

Consensus and Cooperation in Networked Multi-Agent Systems (2007, background): “This includes subjects such as consensus [47, 9, 5, 15, 54, 8, 79], collective behavior of flocks and swarms [66, 80, 60, 95, 30], sensor fusion [64, 71, 33], random networks [34, 73], synchronization of coupled oscillators [81, 39 , 72, 73, 14], algebraic connectivity6of complex networks [65, 12, ”
On the Critical Coupling for Kuramoto Oscillators (2010, unknown): “1) can be equivalently stated as K > (ωmax−ωmin)/ sin(γ) and thus improves the sufficient bounds [15, 31, 17, 46, 20, 26].”
Convergence Speed of Unsteady Distributed Consensus: Decay Estimate Along the Settling Spanning-Trees (2006, background): “Questions of this nature arise in peer-to-peer and sensors networks [7, 1], in the manoeuvring of groups of vehicles [2, 22, 33], in the study of TCP protocols [4], in the theory of coupled oscillators [15, 24, 20, 30], in neural networks [19], but also in apparently distant fields such as in the st”
Communication‐based performance bounds in nonlinear coordinated control (2011, methodology): “The Kuramoto algorithm in particular is known to have exponential convergence near its equilibrium points [4–6], and Kuramoto oscillator consensus has been studied comprehensively for many years [7, 8].”
On Exponential Synchronization of Kuramoto Oscillators (2009, methodology): “The oscillators are said to synchronize if lim t→∞ | ˙ θi(t) − ˙ θj(t)| = 0 ∀i, j = 1, . . . , N Recently, control theoretic methods have been used in [2], [3], [7], [8], [14], [15], [16], [21], [19] to address the synchronization phenomenon.”
Algebraic Connectivity Ratio of Ramanujan Graphs (2007, background): “Some problems in which algebraic connectivity plays a central role include consensus problems [23], [32], [26], [11], [29], [4], [5], [21], synchronization of coupled oscillators [30], [12], [25], flocking and swarming [20], [9], belief propagation and distributed hypothesis testing [27], [22], and ”
A survey of consensus problems in multi-agent coordination (2005, background): “In [47], the authors analyze the stability of classic Kuramoto model of coupled nonlinear oscillators.”
Graph theory and networks in Biology. (2006, background): “The majority of these results only strictly hold in the thermodynamic limit when N → ∞, although some results are available for large but finite populations [204] and in general finite-dimensional models are beginning to receive more attention [196].”
Distributed Coordination Control of Multiagent Systems While Preserving Connectedness (2007, methodology): “Moreover, similar Lyapunov function arguments were employed in [34], where the stability of coupled nonlinear oscillator networks was investigated.”
Algebraic Connectivity of Interdependent Networks (2014, background): “To date, the problem of how the structural properties of a network influences the convergence and stability of its synchronized states has been extensively investigated and discussed, both numerically and theoretically [5, 6, 7, 8, 9], with special attention given to networks of coupled oscillators ”
Synchronization in complex networks of phase oscillators: A survey (2014, methodology): “5.2 Consensus, Contraction, & Convexity The interest of the control community in oscillator networks (1) was initially sparked by Jadbabaie et al. (2004) and Moreau (2005), who analyzed networks of identical oscillators as nonlinear extensions of the consensus protocol (5).”
Synchronization of coupled harmonic oscillators in a dynamic proximity network (2009, methodology): “The Kuramoto model was later modified for the scenarios of nearest neighbor interaction [ 8 ], switching topologies and presence of nonhomogeneous delays [9].”
Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators (2009, methodology): “Note also that the Lyapunov function W(θ) reduces to the one used in [23], [24] and can be used to prove [23, Theorem 4.”
Stability of multiagent systems with time-dependent communication links (2005, unknown): “Denote the state of oscillator by and consider for each communication graph the Kuramoto equation [8], [9]”
Stabilization of Planar Collective Motion With Limited Communication (2008, background): “An alternative proof is provided in [21].”