Abstract
In this article, we analyze a nonlocal ring network of adaptively coupled phase oscillators. We observe a variety of frequency-synchronized states such as phase-locked, multicluster and solitary states. For an important subclass of the phase-locked solutions, the rotating waves, we provide a rigorous stability analysis. This analysis shows a strong dependence of their stability on the coupling structure and the wavenumber which is a remarkable difference to an all-to-all coupled network. Despite the fact that solitary states have been observed in a plethora of dynamical systems, the mechanisms behind their emergence were largely unaddressed in the literature. Here, we show how solitary states emerge due to the adaptive feature of the network and classify several bifurcation scenarios in which these states are created and stabilized.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. Gerstner, R. Kempter, J.L. von Hemmen, H. Wagner, Nature 383, 76 (1996)
C. Clopath, L. Büsing, E. Vasilaki, W. Gerstner, Nat. Neurosci. 13, 344 (2010)
H. Markram, J. Lübke, M. Frotscher, B. Sakmann, Science 275, 213 (1997)
L.F. Abbott, S. Nelson, Nat. Neurosci. 3, 1178 (2000)
N. Caporale, Y. Dan, Ann. Rev. Neurosci. 31, 25 (2008)
O.V. Popovych, S. Yanchuk, P. Tass, Sci. Rep. 3, 2926 (2013)
L. Lücken, O.V. Popovych, P. Tass, S. Yanchuk, Phys. Rev. E 93, 032210 (2016)
S. Jain, S. Krishna, Proc. Natl. Acad. Sci. 98, 543 (2001)
T. Gross, B. Blasius, J. R. Soc. Interface 5, 259 (2008)
F. Pasemann, Neural Netw. 8, 421 (1995)
P.C. Bressloff, S. Coombes, B. de Souza, Phys. Rev. Lett. 79, 2791 (1997)
S. Yanchuk, M. Wolfrum, Phys. Rev. E 77, 26212 (2008)
M. Bonnin, Physica D 238, 77 (2009)
W. Zou, M. Zhan, SIAM J. Appl. Dyn. Syst. 8, 1324 (2009)
Y. Horikawa, H. Kitajima, Physica D 238, 216 (2009)
P. Perlikowski, S. Yanchuk, O.V. Popovych, P. Tass, Phys. Rev. E 82, 036208 (2010)
I. Omelchenko, Y. Maistrenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 106, 234102 (2011)
M. Kantner, S. Yanchuk, Philos. Trans. R. Soc. A 371, 20120470 (2013)
I. Omelchenko, O.E. Omel’chenko, P. Hövel, E. Schöll, Phys. Rev. Lett. 110, 224101 (2013)
S. Yanchuk, P. Perlikowski, M. Wolfrum, A. Stefanski, T. Kapitaniak, Chaos 25, 033113 (2015)
V. Klinshov, D. Shchapin, S. Yanchuk, M. Wolfrum, O. D’Huys, V.I. Nekorkin, Phys. Rev. E 96, 042217 (2017)
O.E. Omel’chenko, Nonlinearity 31, R121 (2018)
A. Compte, M.V. Sanchez-Vives, D.A. McCormick, X.J. Wang, J. Neurophysiol. 89, 2707 (2003)
O. Sporns,Networks of the brain (MIT Press, Cambridge, MA, 2011)
O.V. Popovych, S. Yanchuk, P. Tass, Phys. Rev. Lett. 107, 228102 (2011)
H. Sakaguchi, Y. Kuramoto, Prog. Theor. Phys. 76, 576 (1986)
A. Pikovsky, M. Rosenblum, Chaos 25, 097616 (2015)
Q. Ren, J. Zhao, Phys. Rev. E 76, 016207 (2007)
T. Aoki, T. Aoyagi, Phys. Rev. Lett. 102, 034101 (2009)
T. Aoki, T. Aoyagi, Phys. Rev. E 84, 066109 (2011)
C.B. Picallo, H. Riecke, Phys. Rev. E 83, 036206 (2011)
L. Timms, L.Q. English, Phys. Rev. E 89, 032906 (2014)
V.I. Nekorkin, D.V. Kasatkin, AIP Conf. Proc. 1738, 210010 (2016)
D.V. Kasatkin, S. Yanchuk, E. Schöll, V.I. Nekorkin, Phys. Rev. E 96, 062211 (2017)
V. Avalos-Gaytán, J.A. Almendral, I. Leyva, F. Battiston, V. Nicosia, V. Latora, S. Boccaletti, Phys. Rev. E 97, 042301 (2018)
R. Berner, E. Schöll, S. Yanchuk, SIAM J. Appl. Dyn. Syst. 18, 2227 (2019)
R. Berner, J. Fialkowski, D.V. Kasatkin, V.I. Nekorkin, S. Yanchuk, E. Schöll, Chaos 29, 103134 (2019)
D.V. Kasatkin, V.I. Nekorkin, Eur. Phys. J. Special Topics 227, 1051 (2018)
O.V. Popovych, M.N. Xenakis, P.A. Tass, PLoS One 10, e0117205 (2015)
S. Chakravartula, P. Indic, B. Sundaram, T. Killingback, PLoS One 12, e0178975 (2017)
V. Röhr, R. Berner, E.L. Lameu, O.V. Popovych, S. Yanchuk, PLoS One 14, e0225094 (2019)
D.S. Bassett, E. Bullmore, B.A. Verchinski, V.S. Mattay, D.R. Weinberger, A. Meyer-Lindenberg, J. Neurosci. 28, 9239 (2008)
D. Meunier, R. Lambiotte, E.T. Bullmore, Front. Neurosci. 4, 200 (2010)
D.S. Bassett, D.L. Greenfield, A. Meyer-Lindenberg, D.R. Weinberger, S.W. Moore, E.T. Bullmore, PLoS Comput Biol. 6, e1000748 (2010)
C. Lohse, D.S. Bassett, K.O. Lim, J.M. Carlson, PLoS Comput. Biol. 10, e1003712 (2014)
R.F. Betzel, J.D. Medaglia, L. Papadopoulos, G.L. Baum, R. Gur, D. Roalf, T.D. Satterthwaite, D.S. Bassett, Netw. Neurosci. 1, 42 (2017)
A. Ashourvan, Q.K. Telesford, T. Verstynen, J.M. Vettel, D.S. Bassett, PLoS One 14, e0215520 (2019)
Y. Maistrenko, B. Penkovsky, M. Rosenblum, Phys. Rev. E 89, 060901 (2014)
H. Wu, M. Dhamala, Phys. Rev. E 98, 032221 (2018)
E. Teichmann, M. Rosenblum, Chaos 29, 093124 (2019)
B. Chen, J.R. Engelbrecht, R.E. Mirollo, Chaos 29, 013126 (2019)
P. Jaros, Y. Maistrenko, T. Kapitaniak, Phys. Rev. E 91, 022907 (2015)
P. Jaros, S. Brezetsky, R. Levchenko, D. Dudkowski, T. Kapitaniak, Y. Maistrenko, Chaos 28, 011103 (2018)
H. Taher, S. Olmi, E. Schöll, Phys. Rev. E 100, 062306 (2019)
K. Sathiyadevi, V.K. Chandrasekar, D.V. Senthilkumar, M. Lakshmanan, J. Phys. A: Math. Theor. 52, 184001 (2019)
E. Rybalova, V.S. Anishchenko, G.I. Strelkova, A. Zakharova, Chaos 29, 071106 (2019)
L. Schülen, S. Ghosh, A.D. Kachhvah, A. Zakharova, S. Jalan, Chaos Solitons Fractals 128, 290 (2019)
M.A. Zaks, P. Tomov, Phys. Rev. E 93, 020201 (2016)
E. Rybalova, N. Semenova, G. Strelkova, V. Anishchenko, Eur. Phys. J. Special Topics 226, 1857 (2017)
T. Kapitaniak, P. Kuzma, J. Wojewoda, K. Czolczynski, Y. Maistrenko, Sci. Rep. 4, 6379 (2014)
V. Semenov, A. Zakharova, Y. Maistrenko, E. Schöll, Europhys. Lett. 115, 10005 (2016)
M. Mikhaylenko, L. Ramlow, S. Jalan, A. Zakharova, Chaos 29, 023122 (2019)
N. Semenova, T. Vadivasova, V. Anishchenko, Eur. Phys. J. Special Topics 227, 1173 (2018)
R.M. Gray, Found. Trends Commun. Inf. Theory 2, 155 (2006)
B. Korte, J. Vygen,Combinatorial Optimization (Springer, Berlin, 2018)
P. Ashwin, O. Burylko, Chaos 25, 013106 (2015)
J. Wojewoda, K. Czolczynski, Y. Maistrenko, T. Kapitaniak, Sci. Rep. 6, 34329 (2016)
K. Premalatha, V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, Phys. Rev. E 94, 012311 (2016)
Y. Maistrenko, S. Brezetsky, P. Jaros, R. Levchenko, T. Kapitaniak, Phys. Rev. E 95, 010203(R) (2017)
P. Ashwin, O. Burylko, Y. Maistrenko, Physica D 237, 454 (2008)
M. Golubitsky, I. Stewart,Singularities and Groups in Bifurcation Theory. Volume 2, Applied Mathematical Sciences (Springer-Verlag, New York, 1988), Vol. 69
Y. Maistrenko, A. Vasylenko, O. Sudakov, R. Levchenko, V.L. Maistrenko, Int. J. Bifurc. Chaos 24, 1440014 (2014)
J. Liesen, V. Mehrmann,Linear Algebra (Springer, Cham, 2015)
Acknowledgments
Open access funding provided by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Berner, R., Polanska, A., Schöll, E. et al. Solitary states in adaptive nonlocal oscillator networks. Eur. Phys. J. Spec. Top. 229, 2183–2203 (2020). https://doi.org/10.1140/epjst/e2020-900253-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2020-900253-0