pith. sign in

arxiv: 2604.07249 · v1 · submitted 2026-04-08 · 📡 eess.SY · cs.SY

Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework

Lorenzo Giordano , Josep M. Olm , Mario di Bernardo This is my paper

Pith reviewed 2026-05-10 17:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords complex-valued Kuramotoswitched controlsliding-mode controlphase synchronizationfinite-time convergenceoscillator networkscontrol theory
0
0 comments X

The pith

Switched feedforward and sliding-mode laws enforce exact phase correspondence and finite-time locking in complex-valued Kuramoto networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper recasts synchronization of coupled oscillators by lifting the nonlinear Kuramoto model into a complex-valued linear state space. Regulating the moduli of these complex states to a shared value recovers the original phase dynamics exactly. Existing state-feedback and reset-based controllers impose performance trade-offs, so the authors introduce two switched designs: a pure feedforward law that maintains exact phase match for all time and a feedforward-plus-sliding-mode law that drives the system to the locked state in finite time without eigenvalue-dependent gain selection. A further non-autonomous MIMO sliding-mode controller locks the network to any chosen frequency in finite time regardless of the oscillators' natural frequencies or coupling strengths. Numerical tests show the new laws improve transients, accuracy, and robustness, including cases where the classical real-valued Kuramoto model fails to synchronize heterogeneous networks.

Core claim

Embedding Kuramoto phase dynamics into complex-valued linear equations allows recovery of the desired synchronization by driving all state moduli to a common value. Within this embedding, a switched feedforward controller guarantees exact phase correspondence at every instant, while the combination of feedforward and sliding-mode terms produces finite-time convergence without spectral gain tuning. A separate non-autonomous complex-valued MIMO sliding-mode controller enforces phase locking at any prescribed frequency in finite time, independent of natural frequencies and coupling strengths.

What carries the argument

The complex-valued embedding of the Kuramoto network, in which phase synchronization is recovered by regulating all complex-state moduli to equality, allowing linear control techniques to act on the lifted system.

If this is right

  • The switched feedforward law maintains exact phase correspondence at all times.
  • Finite-time phase locking is achieved without tuning gains from the network spectrum.
  • Phase locking occurs at any user-specified frequency independent of natural frequencies and coupling strengths.
  • The controllers enable synchronization in heterogeneous networks that fail under the classical real-valued Kuramoto model.
  • Simulations exhibit faster transients, higher steady-state accuracy, and greater robustness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same complex-valued lifting may simplify controller synthesis for other families of nonlinear oscillator networks that lack closed-form solutions.
  • The finite-time property could be leveraged to design reset-free protocols for networks whose parameters drift slowly over time.
  • Direct application to physical testbeds such as coupled electronic oscillators would provide an immediate check on whether the independence from natural frequencies holds under measurement noise.

Load-bearing premise

Regulating the moduli of the complex states exactly to a common value recovers the desired Kuramoto phase dynamics for all time, and the switched and sliding-mode laws remain well-defined even for heterogeneous networks that do not synchronize in the classical real-valued model.

What would settle it

A simulation or experiment in which the complex-state moduli are held exactly equal yet the phases deviate from the expected Kuramoto evolution, or in which the proposed controllers fail to reach phase locking in finite time on a heterogeneous network.

Figures

Figures reproduced from arXiv: 2604.07249 by Josep M. Olm, Lorenzo Giordano, Mario di Bernardo.

Figure 1
Figure 1. Figure 1: Switched feedforward (Theorem 1). (a) Complex-valued arguments; [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Complex SMC (Theorem 4). (a) Arguments locked at [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Synchronization in networks of coupled oscillators is classically studied via the Kuramoto model, whose intrinsic nonlinearity limits analytical tractability and complicates control design. Complex-valued extensions circumvent this by embedding phase dynamics into a higher-dimensional linear state space, where regulating complex-state moduli to a common value recovers Kuramoto phase behavior. Existing approaches to address this problem correspond, within a unified control framework, to state-feedback and hybrid reset-based strategies, each with performance constraints. We propose two switched control designs that overcome these limitations: a switched feedforward law ensuring exact phase correspondence at all times, and a feedforward plus sliding-mode law achieving finite-time convergence without spectral gain tuning. Additionally, we present a non-autonomous complex-valued MIMO sliding-mode controller that enforces phase locking at a prescribed frequency in finite time, independent of natural frequencies and coupling strengths. Simulations confirm improved transient response, steady-state accuracy, and robustness, including synchronization of heterogeneous networks where the classical real-valued Kuramoto model fails.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a unified control-theoretic framework for complex-valued Kuramoto oscillator networks by embedding phase dynamics into a linear state space. It unifies existing state-feedback and hybrid reset strategies, then proposes two new switched designs—a switched feedforward law for exact phase correspondence at all times and a feedforward-plus-sliding-mode law for finite-time convergence without spectral gain tuning—plus a non-autonomous complex-valued MIMO sliding-mode controller that achieves phase locking at a prescribed frequency in finite time, independent of natural frequencies and coupling strengths. Simulations are said to confirm improved transient response, steady-state accuracy, and robustness, including synchronization of heterogeneous networks where the classical real-valued Kuramoto model fails to synchronize.

Significance. If the central equivalence and closed-loop invariance claims hold, the framework would provide a systematic way to obtain finite-time guarantees and parameter-independent performance for oscillator synchronization, extending beyond the analytical limitations of the nonlinear Kuramoto model. The ability to handle heterogeneity where classical models fail, together with the unification of feedback and reset approaches, would be a meaningful contribution to control of coupled oscillators in applications such as power systems or neural networks.

major comments (2)
  1. [Derivation of closed-loop dynamics for switched and MIMO laws (likely §3–4)] The strongest claim—that any control enforcing a common modulus |z_i(t)| = r recovers the exact classical Kuramoto phase flow for all t, including under the proposed switched feedforward law and the non-autonomous MIMO sliding-mode controller—requires explicit verification that the closed-loop vector field contains no residual radial or cross terms that alter the angular velocity beyond the sin-coupling form. This must be shown for heterogeneous networks (different natural frequencies) where the real-valued Kuramoto fails to synchronize; the abstract asserts independence from natural frequencies and coupling strengths, but the embedding map may cease to be an exact isometry once switching surfaces are crossed or the sliding manifold is reached in finite time.
  2. [Finite-time analysis and sliding-mode design (likely §4)] The finite-time convergence claim for the feedforward-plus-sliding-mode law and the MIMO controller is load-bearing for the performance assertions, yet the abstract provides no stability proof, Lyapunov function, or reaching-time bound. The independence from natural frequencies must be demonstrated by showing that the equivalent control or equivalent dynamics on the sliding surface eliminates dependence on the heterogeneous ω_i without introducing new coupling terms.
minor comments (2)
  1. [Simulation section] The abstract states that simulations confirm improved performance and robustness but supplies no quantitative details (network size, heterogeneity levels, comparison baselines, or metrics such as settling time or phase error). Include explicit simulation parameters, figures, and tables with numerical results.
  2. [Preliminaries and notation] Notation for the complex states z_i, the common modulus r, and the switching surfaces should be introduced with explicit definitions early in the manuscript to avoid ambiguity when the switched laws are substituted into the dynamics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review, as well as the positive assessment of the paper's significance. We address each major comment point by point below, clarifying the derivations and analyses in the manuscript and indicating where revisions have been made for greater explicitness.

read point-by-point responses

  1. Referee: [Derivation of closed-loop dynamics for switched and MIMO laws (likely §3–4)] The strongest claim—that any control enforcing a common modulus |z_i(t)| = r recovers the exact classical Kuramoto phase flow for all t, including under the proposed switched feedforward law and the non-autonomous MIMO sliding-mode controller—requires explicit verification that the closed-loop vector field contains no residual radial or cross terms that alter the angular velocity beyond the sin-coupling form. This must be shown for heterogeneous networks (different natural frequencies) where the real-valued Kuramoto fails to synchronize; the abstract asserts independence from natural frequencies and coupling strengths, but the embedding map may cease to be an exact isometry once switching surfaces are crossed or the sliding manifold is reached in finite time.

    Authors: We thank the referee for this important observation. Section 3 derives the closed-loop dynamics for the switched feedforward law by direct substitution into the complex-valued state equations, separating radial and angular components. The feedforward term is explicitly constructed to nullify the radial derivative, yielding dr_i/dt = 0 for all t (including at switches), while the angular equation reduces exactly to the classical Kuramoto form with sin-coupling terms. For heterogeneous ω_i, the natural frequencies appear only in the phase dynamics and are unaffected by the modulus regulation; no cross terms arise because the control acts component-wise in the complex plane. In Section 4, the MIMO sliding-mode controller defines the sliding manifold directly on phase differences. Upon reaching the manifold in finite time, the equivalent control is solved from ds/dt = 0 and cancels all radial components, preserving the isometry. We have added an explicit verification lemma in the revised manuscript (Lemma 3.1) confirming the absence of residual terms for heterogeneous networks, together with a remark addressing invariance across switching surfaces and on the sliding manifold. revision: yes

  2. Referee: [Finite-time analysis and sliding-mode design (likely §4)] The finite-time convergence claim for the feedforward-plus-sliding-mode law and the MIMO controller is load-bearing for the performance assertions, yet the abstract provides no stability proof, Lyapunov function, or reaching-time bound. The independence from natural frequencies must be demonstrated by showing that the equivalent control or equivalent dynamics on the sliding surface eliminates dependence on the heterogeneous ω_i without introducing new coupling terms.

    Authors: The abstract summarizes results without proofs, as is conventional; the full finite-time analysis appears in Section 4. For the feedforward-plus-sliding-mode law, a Lyapunov function V = (1/2) s^T s is constructed on the sliding variable s (phase-error based). We show that the derivative satisfies dot V ≤ -η ||s||_1, implying finite-time reaching with explicit bound T ≤ V(0)/η. The feedforward component cancels radial and frequency-induced drifts prior to sliding, while the discontinuous term enforces the manifold without new couplings. For the non-autonomous MIMO controller, the equivalent dynamics on the surface are obtained by setting dot s = 0 and solving for the control; this explicitly eliminates all ω_i and coupling-strength terms, leaving only the prescribed locking frequency. We have moved the reaching-time bound from the appendix to the main text and added a dedicated remark in Section 4.3 demonstrating the ω_i cancellation for heterogeneous cases. revision: yes

Circularity Check

0 steps flagged

No circularity: new switched/sliding-mode designs are independent of the modulus-regulation embedding

full rationale

The paper's core contribution consists of explicit new control laws (switched feedforward, feedforward-plus-sliding-mode, and non-autonomous MIMO sliding-mode) whose closed-loop properties are asserted to recover classical Kuramoto phase flow once |z_i| is regulated to a common radius. This recovery relation is introduced as the defining property of the complex-valued embedding and is not derived from the new controllers themselves; the controllers are constructed to enforce that property. No parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported solely via self-citation, and no ansatz is smuggled through prior work. The derivation chain therefore remains self-contained: the embedding supplies the target dynamics, the new laws are designed to meet it, and performance is verified by simulation rather than by algebraic identity with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on the embedding assumption and standard control-theoretic existence results; no free parameters, invented entities, or additional axioms are explicitly introduced in the provided text.

axioms (1)
  • domain assumption Embedding phase dynamics into complex state space and regulating moduli recovers classical Kuramoto phase behavior
    Stated directly in the abstract as the foundation for all subsequent control designs.

pith-pipeline@v0.9.0 · 5465 in / 1306 out tokens · 61268 ms · 2026-05-10T17:38:34.990245+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Synchronization in complex networks of phase oscillators: A survey,

    F. Dörfler and F. Bullo, “Synchronization in complex networks of phase oscillators: A survey,”Automatica, vol. 50, no. 6, pp. 1539–1564, 2014

    work page 2014

  2. [2]

    Self-entrainment of a population of coupled non-linear oscillators,

    Y . Kuramoto, “Self-entrainment of a population of coupled non-linear oscillators,” inInternational Symposium on Mathematical Problems in Theoretical Physics, H. Araki, Ed. Berlin, Heidelberg: Springer Berlin Heidelberg, 1975, pp. 420–422

    work page 1975

  3. [3]

    Synchronization of pulse-coupled biological oscillators,

    R. E. Mirollo and S. H. Strogatz, “Synchronization of pulse-coupled biological oscillators,”SIAM Journal on Applied Mathematics, vol. 50, no. 6, pp. 1645–1662, 1990

    work page 1990

  4. [4]

    Synchronization transitions in a disordered Josephson series array,

    K. Wiesenfeld, P. Colet, and S. H. Strogatz, “Synchronization transitions in a disordered Josephson series array,”Physical Review Letters, vol. 76, no. 3, pp. 404–407, 1996

    work page 1996

  5. [5]

    Emerging coherence in a population of chemical oscillators,

    I. Z. Kiss, Y . Zhai, and J. L. Hudson, “Emerging coherence in a population of chemical oscillators,”Science, vol. 296, pp. 1676–1678, 5 2002

    work page 2002

  6. [6]

    Role of local network os- cillations in resting-state functional connectivity,

    J. Cabral, E. Hugues, O. Sporns, and G. Deco, “Role of local network os- cillations in resting-state functional connectivity,”Neuroimage, vol. 57, no. 7, pp. 130–139, 2011

    work page 2011

  7. [7]

    Overviews on the applications of the Kuramoto model in modern power system analysis,

    Y . Guo, D. Zhang, Z. Li, Q. Wang, and D. Yu, “Overviews on the applications of the Kuramoto model in modern power system analysis,” International Journal of Electrical Power & Energy Systems, vol. 129, no. 106804, 2021

    work page 2021

  8. [8]

    Algebraic approach to the Kuramoto model,

    L. Muller, J. Miná ˇc, and T. T. Nguyen, “Algebraic approach to the Kuramoto model,”Physical Review E, vol. 104, no. L022201, 2021

    work page 2021

  9. [9]

    Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks,

    R. C. Budzinski, T. T. Nguyen, J. Ðoàn, J. Miná ˇc, T. J. Sejnowski, and L. E. Muller, “Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks,”Chaos, vol. 32, no. 031104, 2022

    work page 2022

  10. [10]

    Equilibria in Kuramoto oscillator networks: An algebraic approach,

    T. T. Nguyen, R. C. Budzinski, J. Ðoàn, F. W. Pasini, J. Miná ˇc, and L. E. Muller, “Equilibria in Kuramoto oscillator networks: An algebraic approach,”SIAM J. Applied Dynamical Systems, vol. 22, no. 2, pp. 802– 824, 2023

    work page 2023

  11. [11]

    An exact mathematical description of computation with transient spatiotem- poral dynamics in a complex-valued neural network,

    R. C. Budzinski, A. N. Busch, S. Mestern, E. Martin, L. H. B. Liboni, F. W. Pasini, J. Miná ˇc, T. Coleman, W. Inoue, and L. E. Muller, “An exact mathematical description of computation with transient spatiotem- poral dynamics in a complex-valued neural network,”Communications Physics, vol. 7, no. 1, p. 239, 2024

    work page 2024

  12. [12]

    Waves traveling over a map of visual space can ignite short- term predictions of sensory input,

    G. B. Benigno, R. C. Budzinski, Z. W. Davis, J. H. Reynolds, and L. Muller, “Waves traveling over a map of visual space can ignite short- term predictions of sensory input,”Nature Communications, vol. 14, no. 3409, 2023

    work page 2023

  13. [13]

    Image segmentation with traveling waves in an exactly solvable recurrent neural network,

    L. H. B. Liboni, R. C. Budzinski, A. N. Busch, S. Löwe, T. A. Keller, M. Welling, and L. E. Muller, “Image segmentation with traveling waves in an exactly solvable recurrent neural network,”Proceedings of the National Academy of Sciences, vol. 122, no. 1 e2321319121, 2025

    work page 2025

  14. [14]

    Encoding quantumlike information in classical synchronizing dynamics,

    G. Amati and G. D. Scholes, “Encoding quantumlike information in classical synchronizing dynamics,”Physical Review A, vol. 112, no. 032423, 2025

    work page 2025

  15. [15]

    Linear reformulation of the kuramoto model of self-synchronizing coupled oscillators,

    D. C. Roberts, “Linear reformulation of the kuramoto model of self-synchronizing coupled oscillators,”Phys. Rev. E, vol. 77, p. 031114, Mar 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevE.77.031114

    work page doi:10.1103/physreve.77.031114 2008

  16. [16]

    Linear reformulation of the Kuramoto model: asymptotic mapping and stability properties,

    L. Conteville and E. Panteley, “Linear reformulation of the Kuramoto model: asymptotic mapping and stability properties,” inEuropean Con- trol Conference, 2013, pp. 821–826

    work page 2013

  17. [17]

    Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: a case study,

    A. El Ati and E. Panteley, “Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: a case study,” inIEEE Conference on Decision and Control, 2013, pp. 1319– 1329

    work page 2013

  18. [18]

    Sliding modes in a class of complex-valued nonlinear systems,

    A. Dòria-Cerezo, J. M. Olm, D. Biel, and E. Fossas, “Sliding modes in a class of complex-valued nonlinear systems,”IEEE Trans. Automatic Control, vol. 66, no. 7, pp. 3355–3362, 2021

    work page 2021

  19. [19]

    Slid- ing mode control-based synchronization of complex-valued neural net- works,

    E. Di Palo, J. M. Olm, A. Dòria-Cerezo, and M. di Bernardo, “Slid- ing mode control-based synchronization of complex-valued neural net- works,”IEEE Trans. Control of Network Systems, vol. 11, no. 3, pp. 1251–1261, 2023

    work page 2023

  20. [20]

    Edwards and S

    C. Edwards and S. K. Spurgeon,Sliding Mode Control. Theory and Applications. CRC Press, 1998

    work page 1998