pith. sign in

arxiv: 2509.18279 · v3 · pith:FKEGSKMJnew · submitted 2025-09-22 · 🌊 nlin.AO · cond-mat.stat-mech

Emergent Topology of Optimal Networks for Synchrony

Guram Mikaberidze , Dane Taylor This is my paper

Pith reviewed 2026-05-18 14:14 UTC · model grok-4.3

classification 🌊 nlin.AO cond-mat.stat-mech
keywords optimal networkssynchronyKuramoto oscillatorsbipartite topologyphase lockingcoupling budgetemergent topologygradient optimization
0
0 comments X

The pith

Gradient-based optimization under a coupling budget produces sparse bipartite networks that globally phase-lock above a critical value with no synchronization threshold.

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

The paper develops a gradient-based method to maximize synchrony in networks of heterogeneous oscillators while keeping the total coupling strength fixed. The resulting networks consistently adopt sparse, bipartite, elongated, and monophilic structures that hold across models from power-grid equations to chaotic Rössler oscillators. For Kuramoto oscillators the authors derive a constructive theory in which a nonlinear equation selects coupled pairs and a variational principle sets the budget per node. This theory proves that global phase-locking occurs once the budget exceeds a calculable critical value, accompanied by critical scaling, rather than requiring a minimum coupling strength to overcome incoherence.

Core claim

Optimal weighted networks for synchrony under a fixed coupling budget are sparse, bipartite, elongated, and extremely monophilic; for Kuramoto oscillators these networks provably lack a synchronization threshold, so that global phase-locking emerges when the budget surpasses a calculable critical value and exhibits critical scaling at the transition.

What carries the argument

Gradient-based optimization framework that identifies synchrony-optimal weighted networks, together with a nonlinear differential equation that selects coupled node pairs and a variational principle that allocates the coupling budget in the Kuramoto case.

If this is right

  • Synchrony can be achieved globally with a minimal total coupling budget once a calculable threshold is crossed.
  • The same sparse bipartite topology appears for both power-grid swing equations and chaotic Rössler systems, indicating model-independent design rules.
  • Technologies that rely on collective phase coherence, such as microgrids or laser arrays, can be engineered with far fewer links than conventional dense coupling.
  • The absence of a synchronization threshold removes the need for a minimum coupling strength to overcome incoherence in these optimized networks.

Where Pith is reading between the lines

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

  • The same optimization procedure could be applied to other collective phenomena such as consensus or epidemic spreading to test whether sparse bipartite structures are generically optimal under budget constraints.
  • Physical realizations in power grids or optical arrays would allow direct measurement of the predicted critical scaling near the locking transition.
  • The monophilic property suggests that optimal networks deliberately segregate dissimilar nodes, which may reduce interference in heterogeneous oscillator hardware.

Load-bearing premise

The gradient-based optimizer under a fixed coupling budget is assumed to converge reliably to networks whose emergent sparse bipartite monophilic topology is independent of the particular dynamical model and initial conditions.

What would settle it

Numerical integration of the Kuramoto equations on the reported optimal networks would fail to show global phase-locking above the predicted critical budget or would exhibit a synchronization threshold instead of the claimed critical scaling.

Figures

Figures reproduced from arXiv: 2509.18279 by Dane Taylor, Guram Mikaberidze.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Designing high-performing networks requires optimizing for functionality while respecting physical, geometric, or budget constraints. Yet, mathematical and computational tools to design such systems remain limited, particularly for collective dynamics arising from heterogeneous dynamical units. Here, we develop a gradient-based optimization framework to identify synchrony-optimal weighted networks under a constrained coupling budget. The resulting networks exhibit counterintuitive properties: they are sparse, bipartite, elongated, and extremely monophilic (i.e., the neighbors of any node are similar to one another while differing from the node itself). These structural patterns persist across dynamical models ranging from the power-grid swing equations to chaotic R\"ossler systems, suggesting broad applicability to coupled oscillator technologies. To gain insight, we develop a "constructive" theory for coupled Kuramoto oscillators: a nonlinear differential equation identifies which pairs of nodes are coupled, while a variational principle prescribes the budget allocated to each node. Dynamics unfolding over optimal networks provably lack a synchronization threshold; instead, as the budget exceeds a calculable critical value, the system globally phase-locks, exhibiting critical scaling at the transition. Together, our findings offer design principles for synchrony-dependent technologies with potential applications ranging from microgrids to laser arrays and quantum oscillators.

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 manuscript develops a gradient-based optimization framework to identify weighted networks maximizing synchrony under a fixed coupling budget constraint. Optimal networks are reported to exhibit sparse, bipartite, elongated, and monophilic topologies that persist across dynamical models (power-grid swing equations to chaotic Rössler oscillators). A constructive theory for Kuramoto oscillators is presented, employing a nonlinear differential equation for edge selection and a variational principle for budget allocation, with the claim that optimal networks provably lack a synchronization threshold and instead exhibit critical scaling above a calculable critical budget.

Significance. If the reported topologies are confirmed to be global optima independent of initialization and model details, the work would supply concrete design principles for synchrony in coupled-oscillator technologies. The constructive Kuramoto theory is a positive feature, as it supplies an analytical route to the absence of a threshold and the associated critical scaling. The overall significance is limited by the absence of convergence guarantees for the non-convex optimization, which underpins the central claims of model-independent emergent structure.

major comments (2)
  1. [§3] §3 (Optimization Framework): The gradient-based procedure for maximizing synchrony under the coupling-budget constraint is central to all reported topologies, yet no convergence analysis, basin-of-attraction study, or systematic multi-initialization tests are provided to establish that the sparse bipartite monophilic structures are global rather than local optima. Without this, the claim that these features persist across models rests on unverified numerical trajectories.
  2. [§5.2] §5.2 (Constructive Kuramoto Theory): The nonlinear differential equation for edge selection together with the variational budget-allocation principle is asserted to yield a provable lack of synchronization threshold and critical scaling above a calculable budget. The derivation should explicitly demonstrate invariance under the precise normalization of the budget constraint and confirm that the critical value is independent of any auxiliary assumptions introduced in the variational step.
minor comments (2)
  1. [Abstract] The definition of 'monophilic' (neighbors similar to one another but different from the focal node) is introduced in the abstract and results but would benefit from an explicit mathematical formulation (e.g., a similarity metric or clustering coefficient) at first use.
  2. [Figures] Figure captions for the network visualizations should include the precise value of the coupling budget used and the color scale for edge weights to allow direct comparison with the theoretical critical budget derived in §5.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation of our results. We address each major comment below and have revised the manuscript to incorporate additional numerical evidence and clarifications in the relevant sections.

read point-by-point responses

  1. Referee: [§3] §3 (Optimization Framework): The gradient-based procedure for maximizing synchrony under the coupling-budget constraint is central to all reported topologies, yet no convergence analysis, basin-of-attraction study, or systematic multi-initialization tests are provided to establish that the sparse bipartite monophilic structures are global rather than local optima. Without this, the claim that these features persist across models rests on unverified numerical trajectories.

    Authors: We agree that the optimization is non-convex and that the original manuscript did not include a systematic study of convergence or multiple initializations, which limits the strength of claims regarding global optimality. To address this, we have added a new subsection in §3 together with an appendix that reports results from 100 independent random initializations for each dynamical model. In the large majority of runs the optimizer converges to networks sharing the same sparse bipartite monophilic structure (with only small quantitative differences in edge weights). We also include a brief discussion of the observed basin of attraction. While these experiments provide numerical support rather than a mathematical convergence guarantee, they directly substantiate the robustness of the reported topologies across models and initial conditions. revision: yes

  2. Referee: [§5.2] §5.2 (Constructive Kuramoto Theory): The nonlinear differential equation for edge selection together with the variational budget-allocation principle is asserted to yield a provable lack of synchronization threshold and critical scaling above a calculable budget. The derivation should explicitly demonstrate invariance under the precise normalization of the budget constraint and confirm that the critical value is independent of any auxiliary assumptions introduced in the variational step.

    Authors: We thank the referee for highlighting the need for an explicit invariance statement. In the revised §5.2 we have inserted a new proposition that demonstrates the homogeneity of both the edge-selection ODE and the variational budget-allocation principle with respect to the total coupling budget. We show that any uniform rescaling of the budget leaves the critical threshold and the associated scaling exponents unchanged; the critical budget is expressed solely in terms of the oscillator natural frequencies and the resulting network topology. Auxiliary scaling factors introduced during the variational derivation are shown to cancel exactly, confirming independence from those choices. The updated derivation is now self-contained with respect to the normalization of the constraint. revision: yes

Circularity Check

0 steps flagged

No significant circularity; optimization and constructive theory are independent

full rationale

The paper develops a gradient-based optimization framework to maximize synchrony under a coupling budget constraint, numerically identifying emergent topologies (sparse, bipartite, elongated, monophilic) that persist across multiple oscillator models. It then introduces a separate constructive theory for Kuramoto oscillators consisting of a nonlinear differential equation for edge selection and a variational principle for budget allocation, from which the absence of a synchronization threshold and critical scaling are derived analytically. These steps are presented as complementary rather than tautological: the optimization provides empirical evidence across models, while the Kuramoto theory offers first-principles insight into the optimal case. No load-bearing self-citations, fitted inputs renamed as predictions, or self-definitional reductions appear in the derivation chain. The central claims rest on external dynamical models and standard optimization techniques without reducing to their own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of gradient-based optimization convergence and the applicability of Kuramoto and other oscillator models; no new particles or forces are introduced.

free parameters (1)
  • coupling budget constraint
    Total coupling strength is fixed as an external constraint but its specific value is treated as a tunable parameter whose critical threshold is calculated rather than fitted post hoc.
axioms (2)
  • domain assumption Gradient-based optimization converges to globally optimal network weights under the budget constraint
    Invoked in the development of the framework that identifies synchrony-optimal networks.
  • domain assumption The monophilic and bipartite structure is independent of the specific oscillator model
    Stated as persisting across power-grid swing equations to chaotic Rössler systems.

pith-pipeline@v0.9.0 · 5743 in / 1464 out tokens · 52670 ms · 2026-05-18T14:14:00.910889+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    optimal network science

    admits a closed-form solution [ 11]. We leverage this analytical gradient to accelerate optimization by around one order of magnitude. The optimized structural and phase configurations closely match the analytic solution derived in Eqs. (11)-(13), as shown in Figs. 5(b) and 5(c). 7 DISCUSSION While the dynamics and phase transitions of lattices and random...

    work page

  2. [2]

    A universal concept in nonlinear sciences.Self, 2(3):10–1017, 2001

    Arkady Pikovsky, Michael Rosenblum, J¨ urgen Kurths, and A Synchronization. A universal concept in nonlinear sciences.Self, 2(3):10–1017, 2001

    work page 2001

  3. [3]

    International symposium on mathe- matical problems in theoretical physics.Lecture Notes in Physics, 30:420, 1975

    Yoshiki Kuramoto. International symposium on mathe- matical problems in theoretical physics.Lecture Notes in Physics, 30:420, 1975

    work page 1975

  4. [4]

    From Kuramoto to Crawford: explor- ing the onset of synchronization in populations of coupled oscillators.Physica D: Nonlinear Phenomena, 143(1-4): 1–20, 2000

    Steven H Strogatz. From Kuramoto to Crawford: explor- ing the onset of synchronization in populations of coupled oscillators.Physica D: Nonlinear Phenomena, 143(1-4): 1–20, 2000

    work page 2000

  5. [5]

    Synchronization in com- plex networks.Physics Reports, 469(3):93–153, 2008

    Alex Arenas, Albert D´ ıaz-Guilera, Jurgen Kurths, Yamir Moreno, and Changsong Zhou. Synchronization in com- plex networks.Physics Reports, 469(3):93–153, 2008

    work page 2008

  6. [6]

    Automatic differentiation in PyTorch

    Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Al- ban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch. 2017

    work page 2017

  7. [7]

    Ricky T. Q. Chen. torchdiffeq, 2018. URL https:// github.com/rtqichen/torchdiffeq

    work page 2018

  8. [8]

    En- tangled networks, synchronization, and optimal network topology.Physical Review Letters, 95(18):188701, 2005

    Luca Donetti, Pablo I Hurtado, and Miguel A Munoz. En- tangled networks, synchronization, and optimal network topology.Physical Review Letters, 95(18):188701, 2005

    work page 2005

  9. [9]

    Synchrony-optimized networks of non- identical kuramoto oscillators.Physics Letters A, 372(15): 2618–2622, 2008

    Markus Brede. Synchrony-optimized networks of non- identical kuramoto oscillators.Physics Letters A, 372(15): 2618–2622, 2008

    work page 2008

  10. [10]

    Competitive influence maximization and enhance- ment of synchronization in populations of non-identical kuramoto oscillators.Scientific Reports, 8(1):702, 2018

    Markus Brede, Massimo Stella, and Alexander C Kalloni- atis. Competitive influence maximization and enhance- ment of synchronization in populations of non-identical kuramoto oscillators.Scientific Reports, 8(1):702, 2018

    work page 2018

  11. [11]

    Optimal synchronization of complex networks.Physical Review Letters, 113(14):144101, 2014

    Per Sebastian Skardal, Dane Taylor, and Jie Sun. Optimal synchronization of complex networks.Physical Review Letters, 113(14):144101, 2014

    work page 2014

  12. [12]

    Dane Taylor, Per Sebastian Skardal, and Jie Sun. Syn- chronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the syn- chrony alignment function.SIAM Journal on Applied Mathematics, 76(5):1984–2008, 2016

    work page 1984

  13. [13]

    Grass-roots optimization of coupled oscillator networks.Physical Review E, 106(3):034202, 2022

    Pranick R Chamlagai, Dane Taylor, and Per Sebastian Skardal. Grass-roots optimization of coupled oscillator networks.Physical Review E, 106(3):034202, 2022

    work page 2022

  14. [14]

    Master stability functions for synchronized coupled systems.Physical Review Letters, 80(10):2109, 1998

    Louis M Pecora and Thomas L Carroll. Master stability functions for synchronized coupled systems.Physical Review Letters, 80(10):2109, 1998

    work page 1998

  15. [15]

    Synchronization in small-world systems.Physical Review Letters, 89(5): 054101, 2002

    Mauricio Barahona and Louis M Pecora. Synchronization in small-world systems.Physical Review Letters, 89(5): 054101, 2002

    work page 2002

  16. [16]

    Heterogeneity in oscilla- tor networks: Are smaller worlds easier to synchronize? Physical Review Letters, 91(1):014101, 2003

    Takashi Nishikawa, Adilson E Motter, Ying-Cheng Lai, and Frank C Hoppensteadt. Heterogeneity in oscilla- tor networks: Are smaller worlds easier to synchronize? Physical Review Letters, 91(1):014101, 2003

    work page 2003

  17. [17]

    Synchronization is optimal in nondiagonalizable networks.Physical Review E, 73(6):065106, 2006

    Takashi Nishikawa and Adilson E Motter. Synchronization is optimal in nondiagonalizable networks.Physical Review E, 73(6):065106, 2006

    work page 2006

  18. [18]

    Takashi Nishikawa and Adilson E Motter. Network syn- chronization landscape reveals compensatory structures, quantization, and the positive effect of negative interac- tions.Proceedings of the National Academy of Sciences, 107(23):10342–10347, 2010

    work page 2010

  19. [19]

    Adaptive syn- chronization of dynamics on evolving complex networks

    Francesco Sorrentino and Edward Ott. Adaptive syn- chronization of dynamics on evolving complex networks. Physical Review Letters, 100(11):114101, 2008

    work page 2008

  20. [20]

    On the topology of synchrony optimized networks of a kuramoto-model with non-identical oscillators.Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(2), 2011

    David Kelly and Georg A Gottwald. On the topology of synchrony optimized networks of a kuramoto-model with non-identical oscillators.Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(2), 2011

    work page 2011

  21. [21]

    Ricci, M

    Matthew Ricci, Minju Jung, Yuwei Zhang, Mathieu Chalvidal, Aneri Soni, and Thomas Serre. Kuranet: sys- tems of coupled oscillators that learn to synchronize.arXiv preprint arXiv:2105.02838, 2021

    work page arXiv 2021

  22. [22]

    Consensus and cooperation in networked multi-agent sys- tems.Proceedings of the IEEE, 95(1):215–233, 2007

    Reza Olfati-Saber, J Alex Fax, and Richard M Murray. Consensus and cooperation in networked multi-agent sys- tems.Proceedings of the IEEE, 95(1):215–233, 2007

    work page 2007

  23. [23]

    Tuto- rial on dynamic average consensus: The problem, its applications, and the algorithms.IEEE Control Systems Magazine, 39(3):40–72, 2019

    Solmaz S Kia, Bryan Van Scoy, Jorge Cortes, Randy A Freeman, Kevin M Lynch, and Sonia Martinez. Tuto- rial on dynamic average consensus: The problem, its applications, and the algorithms.IEEE Control Systems Magazine, 39(3):40–72, 2019

    work page 2019

  24. [24]

    External bias and opin- ion clustering in cooperative networks.Automatica, 175: 112224, 2025

    Akshay Nagesh Kamthe, Vishnudatta Thota, Aashi Shri- 13 nate, and Twinkle Tripathy. External bias and opin- ion clustering in cooperative networks.Automatica, 175: 112224, 2025

    work page 2025

  25. [25]

    Monophily in social networks introduces similarity among friends-of- friends.Nature Human Behaviour, 2(4):284–290, 2018

    Kristen M Altenburger and Johan Ugander. Monophily in social networks introduces similarity among friends-of- friends.Nature Human Behaviour, 2(4):284–290, 2018

    work page 2018

  26. [26]

    The paradox of second-order homophily in networks.Scientific Reports, 11(1):13360, 2021

    Anna Evtushenko and Jon Kleinberg. The paradox of second-order homophily in networks.Scientific Reports, 11(1):13360, 2021

    work page 2021

  27. [27]

    Network optimization for syn- chrony, 2025

    Guram Mikaberidze. Network optimization for syn- chrony, 2025. URL https://gitlab.com/mikaberidze/ network-optimization-for-synchrony . GitLab reposi- tory

    work page 2025

  28. [28]

    Onset of synchronization in large networks of coupled oscilla- tors.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 71(3):036151, 2005

    Juan G Restrepo, Edward Ott, and Brian R Hunt. Onset of synchronization in large networks of coupled oscilla- tors.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 71(3):036151, 2005

    work page 2005

  29. [29]

    Synchronization on small-world networks.Physical Review E, 65(2):026139, 2002

    Hyunsuk Hong, Moo-Young Choi, and Beom Jun Kim. Synchronization on small-world networks.Physical Review E, 65(2):026139, 2002

    work page 2002

  30. [30]

    Relations between average distance, heterogeneity and network synchronizability

    Ming Zhao, Tao Zhou, Bing-Hong Wang, Gang Yan, Hui- Jie Yang, and Wen-Jie Bai. Relations between average distance, heterogeneity and network synchronizability. Physica A: Statistical Mechanics and its Applications, 371 (2):773–780, 2006

    work page 2006

  31. [31]

    Do small worlds synchronize fastest?Eu- rophysics Letters, 90(4):48002, 2010

    Carsten Grabow, Steven M Hill, Stefan Grosskinsky, and Marc Timme. Do small worlds synchronize fastest?Eu- rophysics Letters, 90(4):48002, 2010

    work page 2010

  32. [32]

    Topological data analysis of contagion maps for examining spreading processes on networks.Nature Communications, 6(1):7723, 2015

    Dane Taylor, Florian Klimm, Heather A Harrington, Miroslav Kram´ ar, Konstantin Mischaikow, Mason A Porter, and Peter J Mucha. Topological data analysis of contagion maps for examining spreading processes on networks.Nature Communications, 6(1):7723, 2015

    work page 2015

  33. [33]

    Optimal synchroniza- tion of kuramoto oscillators: A dimensional reduction approach.Physical Review E, 92(6):062801, 2015

    Rafael S Pinto and Alberto Saa. Optimal synchroniza- tion of kuramoto oscillators: A dimensional reduction approach.Physical Review E, 92(6):062801, 2015

    work page 2015

  34. [34]

    Lia Papadopoulos, Jason Z Kim, J¨ urgen Kurths, and Danielle S Bassett. Development of structural correlations and synchronization from adaptive rewiring in networks of kuramoto oscillators.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(7), 2017

    work page 2017

  35. [35]

    Braess’s paradox in oscillator networks, desynchronization and power outage

    Dirk Witthaut and Marc Timme. Braess’s paradox in oscillator networks, desynchronization and power outage. New journal of physics, 14(8):083036, 2012

    work page 2012

  36. [36]

    Spontaneous synchrony in power-grid networks.Nature Physics, 9(3):191–197, 2013

    Adilson E Motter, Seth A Myers, Marian Anghel, and Takashi Nishikawa. Spontaneous synchrony in power-grid networks.Nature Physics, 9(3):191–197, 2013

    work page 2013

  37. [37]

    En- hancing synchronization stability in a multi-area power grid.Scientific Reports, 6(1):26596, 2016

    Bing Wang, Hideyuki Suzuki, and Kazuyuki Aihara. En- hancing synchronization stability in a multi-area power grid.Scientific Reports, 6(1):26596, 2016

    work page 2016

  38. [38]

    Understanding braess’ para- dox in power grids.Nature Communications, 13(1):5396, 2022

    Benjamin Sch¨ afer, Thiemo Pesch, Debsankha Manik, Ju- lian Gollenstede, Guosong Lin, Hans-Peter Beck, Dirk Witthaut, and Marc Timme. Understanding braess’ para- dox in power grids.Nature Communications, 13(1):5396, 2022

    work page 2022

  39. [39]

    Vowel recognition with four coupled spin- torque nano-oscillators.Nature, 563(7730):230–234, 2018

    Miguel Romera, Philippe Talatchian, Sumito Tsunegi, Flavio Abreu Araujo, Vincent Cros, Paolo Bortolotti, Juan Trastoy, Kay Yakushiji, Akio Fukushima, Hitoshi Kubota, et al. Vowel recognition with four coupled spin- torque nano-oscillators.Nature, 563(7730):230–234, 2018

    work page 2018

  40. [40]

    Binding events through the mutual synchronization of spintronic nano-neurons.Nature Communications, 13 (1):883, 2022

    Miguel Romera, Philippe Talatchian, Sumito Tsunegi, Kay Yakushiji, Akio Fukushima, Hitoshi Kubota, Shinji Yuasa, Vincent Cros, Paolo Bortolotti, Maxence Ernoult, et al. Binding events through the mutual synchronization of spintronic nano-neurons.Nature Communications, 13 (1):883, 2022

    work page 2022

  41. [41]

    Neuromorphic computing with nanoscale spintronic oscillators.Nature, 547(7664):428–431, 2017

    Jacob Torrejon, Mathieu Riou, Flavio Abreu Araujo, Sum- ito Tsunegi, Guru Khalsa, Damien Querlioz, Paolo Bor- tolotti, Vincent Cros, Kay Yakushiji, Akio Fukushima, et al. Neuromorphic computing with nanoscale spintronic oscillators.Nature, 547(7664):428–431, 2017

    work page 2017

  42. [42]

    Frequency stabilization in nonlinear micromechanical os- cillators.Nature Communications, 3(1):806, 2012

    Dario Antonio, Dami´ an H Zanette, and Daniel L´ opez. Frequency stabilization in nonlinear micromechanical os- cillators.Nature Communications, 3(1):806, 2012

    work page 2012

  43. [43]

    Intrinsic fluctuations and a phase transi- tion in a class of large populations of interacting oscillators

    Hiroaki Daido. Intrinsic fluctuations and a phase transi- tion in a class of large populations of interacting oscillators. Journal of Statistical Physics, 60(5):753–800, 1990

    work page 1990

  44. [44]

    Cluster synchrony in systems of coupled phase oscillators with higher-order coupling.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 84(3): 036208, 2011

    Per Sebastian Skardal, Edward Ott, and Juan G Re- strepo. Cluster synchrony in systems of coupled phase oscillators with higher-order coupling.Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 84(3): 036208, 2011

    work page 2011

  45. [45]

    Multiplicity of singular synchronous states in the kuramoto model of coupled oscillators.Physical Review Letters, 111(20): 204101, 2013

    Maxim Komarov and Arkady Pikovsky. Multiplicity of singular synchronous states in the kuramoto model of coupled oscillators.Physical Review Letters, 111(20): 204101, 2013

    work page 2013

  46. [46]

    Marc Timme, Theo Geisel, and Fred Wolf. Speed of synchronization in complex networks of neural oscillators: analytic results based on random matrix theory.Chaos: An Interdisciplinary Journal of Nonlinear Science, 16(1), 2006

    work page 2006

  47. [47]

    Collective nonlinear dynamics and self- organization in decentralized power grids.Reviews of Modern Physics, 94(1):015005, 2022

    Dirk Witthaut, Frank Hellmann, J¨ urgen Kurths, Ste- fan Kettemann, Hildegard Meyer-Ortmanns, and Marc Timme. Collective nonlinear dynamics and self- organization in decentralized power grids.Reviews of Modern Physics, 94(1):015005, 2022

    work page 2022

  48. [48]

    Impact of network topology on synchrony of oscil- latory power grids.Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(1), 2014

    Martin Rohden, Andreas Sorge, Dirk Witthaut, and Marc Timme. Impact of network topology on synchrony of oscil- latory power grids.Chaos: An Interdisciplinary Journal of Nonlinear Science, 24(1), 2014

    work page 2014

  49. [49]

    The size of the sync basin.Chaos: An Interdisciplinary Journal of Nonlinear Science, 16(1), 2006

    Daniel A Wiley, Steven H Strogatz, and Michelle Girvan. The size of the sync basin.Chaos: An Interdisciplinary Journal of Nonlinear Science, 16(1), 2006

    work page 2006

  50. [50]

    How basin stability complements the linear-stability paradigm.Nature Physics, 9(2):89–92, 2013

    Peter J Menck, Jobst Heitzig, Norbert Marwan, and J¨ urgen Kurths. How basin stability complements the linear-stability paradigm.Nature Physics, 9(2):89–92, 2013

    work page 2013

  51. [51]

    Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equa- tions.Advances in Neural Information Processing Systems, 2018

    work page 2018

  52. [52]

    Springer Science & Business Media, 2012

    Mark H Holmes.Introduction to perturbation methods, volume 20. Springer Science & Business Media, 2012

    work page 2012

  53. [53]

    The calculus of variations.University of Minnesota, 40, 2020

    Jeff Calder. The calculus of variations.University of Minnesota, 40, 2020

    work page 2020