Symmetry-Independent Stability Analysis of Synchronization Patterns
Abstract
The field of network synchronization has seen tremendous growth following the introduction of the master stability function (MSF) formalism, which enables the efficient stability analysis of synchronization in large oscillator networks. However, to make further progress we must overcome the limitations of this celebrated formalism, which focuses on global synchronization and requires both the oscillators and their interaction functions to be identical, while many systems of interest are inherently heterogeneous and exhibit complex synchronization patterns. Here, we establish a generalization of the MSF formalism that can characterize the stability of any cluster synchronization pattern, even when the oscillators and/or their interaction functions are nonidentical. The new framework is based on finding the finest simultaneous block diagonalization of matrices in the variational equation and does not rely on information about network symmetry. This leads to an algorithm that is error-tolerant and orders of magnitude faster than existing symmetry-based algorithms. As an application, we rigorously characterize the stability of chimera states in networks with multiple types of interactions.
Keywords
MSC codes
Formats available
You can view the full content in the following formats:
References
1.
D. M. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008), art. 084103.
2.
D. M. Abrams, L. M. Pecora, and A. E. Motter, Introduction to focus issue: Patterns of network synchronization, Chaos, 26 (2016), art. 094601.
3.
D. M. Abrams and S. H. Strogatz, Chimera states for coupled oscillators, Phys. Rev. Lett., 93 (2004), art. 174102.
4.
M. Aguiar, P. Ashwin, A. Dias, and M. Field, Dynamics of coupled cell networks: Synchrony, heteroclinic cycles and inflation, J. Nonlinear Sci., 21 (2011), pp. 271--323.
5.
A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), pp. 93--153.
6.
P. Ashwin and O. Burylko, Weak chimeras in minimal networks of coupled phase oscillators, Chaos, 25 (2015), art. 013106.
7.
I. Belykh, E. de Lange, and M. Hasler, Synchronization of bursting neurons: What matters in the network topology, Phys. Rev. Lett., 94 (2005), art. 188101.
8.
V. N. Belykh, I. V. Belykh, and E. Mosekilde, Cluster synchronization modes in an ensemble of coupled chaotic oscillators, Phys. Rev. E, 63 (2001), art. 036216.
9.
V. N. Belykh, G. V. Osipov, S. Petrov, J. A. Suykens, and J. Vandewalle, Cluster synchronization in oscillatory networks, Chaos, 18 (2008), art. 037106.
10.
C. Bick, M. Sebek, and I. Z. Kiss, Robust weak chimeras in oscillator networks with delayed linear and quadratic interactions, Phys. Rev. Lett., 119 (2017), art. 168301.
11.
K. A. Blaha, K. Huang, F. Della Rossa, L. Pecora, M. Hossein-Zadeh, and F. Sorrentino, Cluster synchronization in multilayer networks: A fully analog experiment with LC oscillators with physically dissimilar coupling, Phys. Rev. Lett., 122 (2019), art. 014101.
12.
S. Boccaletti, G. Bianconi, R. Criado, C. I. Del Genio, J. Gómez-Gardenes, M. Romance, I. Sendina-Nadal, Z. Wang, and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), pp. 1--122.
13.
S. Boccaletti, J. Kurths, G. Osipov, D. Valladares, and C. Zhou, The synchronization of chaotic systems, Phys. Rep., 366 (2002), pp. 1--101.
14.
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), pp. 175--308.
15.
Y. Cao, W. Yu, W. Ren, and G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Trans. Ind. Informat., 9 (2013), pp. 427--438.
16.
Y. S. Cho, T. Nishikawa, and A. E. Motter, Stable chimeras and independently synchronizable clusters, Phys. Rev. Lett., 119 (2017), art. 084101.
17.
T. Dahms, J. Lehnert, and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), art. 016202.
18.
C. I. del Genio, J. Gómez-Garden͂es, I. Bonamassa, and S. Boccaletti, Synchronization in networks with multiple interaction layers, Sci. Adv., 2 (2016), art. e1601679.
19.
M. Feki, An adaptive chaos synchronization scheme applied to secure communication, Chaos, Solitons & Fractals, 18 (2003), pp. 141--148.
20.
S. Ghosh, A. Kumar, A. Zakharova, and S. Jalan, Birth and death of chimera: Interplay of delay and multiplexing, EPL, 115 (2016), art. 60005.
21.
M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bull. Amer. Math. Soc., 43 (2006), pp. 305--364.
22.
M. Golubitsky, I. Stewart, and A. Török, Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 78--100, https://doi.org/10.1137/040612634.
23.
A. Hagerstorm, Network Symmetries and Synchronization, https://sourceforge.net/projects/networksym/.
24.
A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, Experimental observation of chimeras in coupled-map lattices, Nat. Phys., 8 (2012), pp. 658--661.
25.
J. D. Hart, K. Bansal, T. E. Murphy, and R. Roy, Experimental observation of chimera and cluster states in a minimal globally coupled network, Chaos, 26 (2016), art. 094801.
26.
J. D. Hart, D. C. Schmadel, T. E. Murphy, and R. Roy, Experiments with arbitrary networks in time-multiplexed delay systems, Chaos, 27 (2017), art. 121103.
27.
D. Irving and F. Sorrentino, Synchronization of dynamical hypernetworks: Dimensionality reduction through simultaneous block-diagonalization of matrices, Phys. Rev. E, 86 (2012), art. 056102.
28.
H. Kamei and P. J. A. Cock, Computation of balanced equivalence relations and their lattice for a coupled cell network, SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 352--382, https://doi.org/10.1137/100819795.
29.
K. Kaneko, Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements, Phys. D, 41 (1990), pp. 137--172.
30.
M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, Multilayer networks, J. Complex Netw., 2 (2014), pp. 203--271.
31.
Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 5 (2002), pp. 380--385.
32.
T.-Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer Science & Business Media, 2013.
33.
Z. Li, Z. Duan, G. Chen, and L. Huang, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circuits Syst. I Reg. Papers, 57 (2010), pp. 213--224.
34.
T. Maehara and K. Murota, A numerical algorithm for block-diagonal decomposition of matrix $*$-algebras with general irreducible components, Jpn. J. Ind. Appl. Math, 27 (2010), pp. 263--293.
35.
T. Maehara and K. Murota, Algorithm for error-controlled simultaneous block-diagonalization of matrices, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 605--620, https://doi.org/10.1137/090779966.
36.
S. Majhi, M. Perc, and D. Ghosh, Chimera states in a multilayer network of coupled and uncoupled neurons, Chaos, 27 (2017), art. 073109.
37.
E. A. Martens, S. Thutupalli, A. Fourrière, and O. Hallatschek, Chimera states in mechanical oscillator networks, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 10563--10567.
38.
A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nat. Phys., 9 (2013), pp. 191--197.
39.
K. Murota, Y. Kanno, M. Kojima, and S. Kojima, A numerical algorithm for block-diagonal decomposition of matrix $*$-algebras with application to semidefinite programming, Jpn. J. Ind. Appl. Math, 27 (2010), pp. 125--160.
40.
M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), pp. 167--256, https://doi.org/10.1137/S003614450342480.
41.
V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, and V. Latora, Remote synchronization reveals network symmetries and functional modules, Phys. Rev. Lett., 110 (2013), art. 174102.
42.
R. Olfati-Saber, A. Fax, and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proc. IEEE, 95 (2007), pp. 215--233.
43.
E. Omel'chenko, Y. L. Maistrenko, and P. A. Tass, Chimera states: The natural link between coherence and incoherence, Phys. Rev. Lett., 100 (2008), art. 044105.
44.
O. E. Omel'chenko, The mathematics behind chimera states, Nonlinearity, 31 (2018), pp. R121--R164.
45.
G. Orosz, Decomposition of nonlinear delayed networks around cluster states with applications to neurodynamics, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1353--1386, https://doi.org/10.1137/130915637.
46.
E. Ott and T. M. Antonsen, Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), art. 037113.
47.
M. J. Panaggio and D. M. Abrams, Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity, 28 (2015), pp. R67--R87.
48.
L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), pp. 821--824.
49.
L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), pp. 2109--2112.
50.
L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Cluster synchronization and isolated desynchronization in complex networks with symmetries, Nat. Commun., 5 (2014), art. 4079.
51.
M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76 (1996), pp. 1804--1807.
52.
D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Schöll, Control of synchronization patterns in neural-like boolean networks, Phys. Rev. Lett., 110 (2013), art. 104102.
53.
M. T. Schaub, N. O'Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), art. 094821.
54.
L. Schmidt and K. Krischer, Clustering as a prerequisite for chimera states in globally coupled systems, Phys. Rev. Lett., 114 (2015), art. 034101.
55.
N. Semenova, A. Zakharova, V. Anishchenko, and E. Schöll, Coherence-resonance chimeras in a network of excitable elements, Phys. Rev. Lett., 117 (2016), art. 014102.
56.
G. C. Sethia and A. Sen, Chimera states: The existence criteria revisited, Phys. Rev. Lett., 112 (2014), art. 144101.
57.
R. Sevilla-Escoboza, I. Sendin͂a-Nadal, I. Leyva, R. Gutiérrez, J. Buldú, and S. Boccaletti, Inter-layer synchronization in multiplex networks of identical layers, Chaos, 26 (2016), art. 065304.
58.
A. B. Siddique, L. Pecora, J. D. Hart, and F. Sorrentino, Symmetry- and input-cluster synchronization in networks, Phys. Rev. E, 97 (2018), art. 042217.
59.
F. Sorrentino, Synchronization of hypernetworks of coupled dynamical systems, New J. Phys., 14 (2012), art. 033035.
60.
F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, Complete characterization of the stability of cluster synchronization in complex dynamical networks, Sci. Adv., 2 (2016), art. e1501737.
61.
W. Stein et al., Sage: Open Source Mathematical Software, 2008, www.sagemath.org.
62.
I. Stewart, M. Golubitsky, and M. Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 609--646, https://doi.org/10.1137/S1111111103419896.
63.
S. H. Strogatz, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), pp. 1--20.
64.
L. Tang, X. Wu, J. Lü, J.-a. Lu, and R. M. D'Souza, Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators, Phys. Rev. E, 99 (2019), art. 012304.
65.
M. Timme, F. Wolf, and T. Geisel, Prevalence of unstable attractors in networks of pulse-coupled oscillators, Phys. Rev. Lett., 89 (2002), art. 154105.
66.
M. Tinkham, Group Theory and Quantum Mechanics, Courier Corporation, 2003.
67.
M. R. Tinsley, S. Nkomo, and K. Showalter, Chimera and phase-cluster states in populations of coupled chemical oscillators, Nat. Phys., 8 (2012), pp. 662--665.
68.
J. F. Totz, J. Rode, M. R. Tinsley, K. Showalter, and H. Engel, Spiral wave chimera states in large populations of coupled chemical oscillators, Nat. Phys., 14 (2018), pp. 282--285.
69.
K. Wiesenfeld, P. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series array, Phys. Rev. Lett., 76 (1996), pp. 404--407.
70.
C. R. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators, Phys. Rev. Lett., 110 (2013), art. 064104.
71.
A. Yeldesbay, A. Pikovsky, and M. Rosenblum, Chimeralike states in an ensemble of globally coupled oscillators, Phys. Rev. Lett., 112 (2014), art. 144103.
72.
Y. Zhang and A. E. Motter, Identical synchronization of nonidentical oscillators: When only birds of different feathers flock together, Nonlinearity, 31 (2018), pp. R1--R23.
Information & Authors
Information
Published In
Copyright
© 2020, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 9 July 2019
Accepted: 3 February 2020
Published online: 3 November 2020
Keywords
MSC codes
Authors
Funding Information
Army Research Office https://doi.org/10.13039/100000183 : W911NF-19-1-0383
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Group Synchronization of Heterogeneous Multiagent System With Hierarchical Structure and Beyond Pairwise InteractionsIEEE Transactions on Systems, Man, and Cybernetics: Systems, Vol. 55, No. 3 | 1 Mar 2025
- Mixed obstacle avoidance in mobile chaotic robots with directional keypads and its non-identical generalized synchronizationNonlinear Dynamics, Vol. 113, No. 3 | 1 October 2024
- The transition to synchronization of networked systemsNature Communications, Vol. 15, No. 1 | 10 June 2024
- Patterns of synchronized clusters in adaptive networksCommunications Physics, Vol. 7, No. 1 | 20 June 2024
- Symmetry breaker governs synchrony patterns in neuronal inspired networksChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 34, No. 11 | 6 November 2024
- Symmetry invariance in nonlinear dynamical complex networksChaos, Solitons & Fractals, Vol. 185 | 1 Aug 2024
- CLUSTER DEPENDENCY AND SYNCHRONIZATION OF MULTILAYER DIRECTED NETWORKS FROM A QUOTIENT GRAPH PERSPECTIVEAdvances in Complex Systems, Vol. 27, No. 04n05 | 2 October 2024
- Synchronization in multiplex networksPhysics Reports, Vol. 1060 | 1 Apr 2024
- Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapsesChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 34, No. 4 | 8 April 2024
- Synchronization of Topological Signals on Simplicial Complexes With Higher-Dimensional SimplicesIEEE Transactions on Network Science and Engineering, Vol. 11, No. 1 | 1 Jan 2024
- Cluster Dependency and Synchronization in Multilayer Directed Complex Networks2023 3rd International Conference on Computer Science, Electronic Information Engineering and Intelligent Control Technology (CEI) | 15 Dec 2023
- Bridging functional and anatomical neural connectivity through cluster synchronizationScientific Reports, Vol. 13, No. 1 | 17 December 2023
- Algebraic Decomposition of Model Predictive Control ProblemsIEEE Control Systems Letters, Vol. 7 | 1 Jan 2023
- Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of MatricesIEEE Open Journal of Control Systems, Vol. 2 | 1 Jan 2023
- Decoupling Optimal Control Problems via Simultaneous Block Diagonalization of Matrices2022 IEEE 61st Conference on Decision and Control (CDC) | 6 Dec 2022
- Multistability and anomalies in oscillator models of lossy power gridsNature Communications, Vol. 13, No. 1 | 6 September 2022
- Pinning control of networks: Dimensionality reduction through simultaneous block-diagonalization of matricesChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 11 | 4 November 2022
- Stability of multiple attractors in the unidirectionally coupled circular networks of limit cycle oscillatorsCommunications in Nonlinear Science and Numerical Simulation, Vol. 111 | 1 Aug 2022
- When multilayer links exchange their roles in synchronizationPhysical Review E, Vol. 106, No. 2 | 26 August 2022
- Forget partitions? Not yet…2022 IEEE International Symposium on Circuits and Systems (ISCAS) | 28 May 2022
- Matryoshka and disjoint cluster synchronization of networksChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 4 | 4 April 2022
- Sinusoidal and nonsinusoidal patterns in amplitude envelope synchronizationPhysical Review E, Vol. 105, No. 4 | 20 April 2022
- Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functionsPhysical Review E, Vol. 105, No. 2 | 4 February 2022
- Symmetry-breaking mechanism for the formation of cluster chimera patternsChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 1 | 3 January 2022
- Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networksPhysical Review E, Vol. 105, No. 1 | 24 January 2022
- One-way dependent clusters and stability of cluster synchronization in directed networksNature Communications, Vol. 12, No. 1 | 1 July 2021
- Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactionsCommunications Physics, Vol. 4, No. 1 | 26 August 2021
- Exponential stability for systems of delay differential equations with block matricesApplied Mathematics Letters, Vol. 121 | 1 Nov 2021
- Cluster synchronization of networks via a canonical transformation for simultaneous block diagonalization of matricesChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 31, No. 11 | 9 November 2021
- Group synchrony, parameter mismatches, and intragroup connectionsPhysical Review E, Vol. 104, No. 5 | 30 November 2021
- Synchronization in Networks With Heterogeneous Adaptation Rules and Applications to Distance-Dependent Synaptic PlasticityFrontiers in Applied Mathematics and Statistics, Vol. 7 | 15 July 2021
- Synchronization scenarios in three-layer networks with a hubChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 31, No. 7 | 15 July 2021
- IntroductionPatterns of Synchrony in Complex Networks of Adaptively Coupled Oscillators | 1 June 2021
- Optimal cost tuning of frustration: Achieving desired states in the Kuramoto-Sakaguchi modelPhysical Review E, Vol. 103, No. 1 | 27 January 2021
- Desynchronization Transitions in Adaptive NetworksPhysical Review Letters, Vol. 126, No. 2 | 15 January 2021
View Options
View options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].