## Chimera states through invariant manifold theory.(English)Zbl 07368123

Y. Kuramoto and D. Battogtokh [Nonlinear Phenom. Complex Syst., 5, No. 4, 380–385 (2002)] observed the coexistence of synchronous and asynchronous oscillations in a ring of identical coupled oscillators. This phenomenon of partial synchronization in networks of identical coupled oscillators is known as “chimera” and it is observed in a wide range of experimental settings.
In this paper the authors obtain some mathematical results on the chimera behavior in a large but finite size network of identical oscillators, with a specific modular structure and a relatively general type of coupling. A simple example is the union of two identical, symmetrically coupled star subnetworks. The key ingredients that underlie the analysis are a dimensional reduction due to a Möbius group symmetry, averaging theory and normally hyperbolic invariant manifold theory.

### MSC:

 37D10 Invariant manifold theory for dynamical systems 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C29 Averaging method for ordinary differential equations 34D06 Synchronization of solutions to ordinary differential equations 34C45 Invariant manifolds for ordinary differential equations 70K50 Bifurcations and instability for nonlinear problems in mechanics

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JiTCODE; JiTCSDE; JiTCDDE
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### References:

 [1] Abrams, D. M.; Mirollo, R.; Strogatz, S. H.; Wiley, D. A., Solvable model for chimera states of coupled oscillators, Phys. Rev. Lett., 101 (2008) [2] Ansmann, G., Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE, Chaos, 28 (2018) · Zbl 1390.34005 [3] Ashwin, P.; Burylko, O., Weak chimeras in minimal networks of coupled phase oscillators, Chaos, 25 (2015) · Zbl 1345.34052 [4] Barabási, A-L; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512 (1999) · Zbl 1226.05223 [5] Bick, C.; Ashwin, P., Chaotic weak chimeras and their persistence in coupled populations of phase oscillators, Nonlinearity, 29, 1468 (2016) · Zbl 1355.37041 [6] Chicone, C., Ordinary Differential Equations with Applications, vol 34 (2006), New York: Springer, New York · Zbl 1120.34001 [7] Eldering, J., Normally Hyperbolic Invariant Manifolds: The Noncompact Case (2013), Paris: Atlantis Press, Paris · Zbl 1303.37011 [8] Fenichel, N., Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21, 193-226 (19711972) · Zbl 0246.58015 [9] Haugland, S. W.; Schmidt, L.; Krischer, K., Self-organized alternating chimera states in oscillatory media, Sci. Rep., 5, 9883 (2015) [10] Hirsch, M. W.; Pugh, C. C.; Shub, M., Invariant Manifolds (1977), Berlin: Springer, Berlin [11] Kemeth, F. P.; Haugland, S. W.; Schmidt, L.; Kevrekidis, I. G.; Krischer, K., A classification scheme for chimera states, Chaos, 26 (2016) [12] Ko, T-W; Ermentrout, G. B., Bistability between synchrony and incoherence in limit-cycle oscillators with coupling strength inhomogeneity, Phys. Rev. E, 78 (2008) [13] Ko, T-W; Ermentrout, G. B., Partially locked states in coupled oscillators due to inhomogeneous coupling, Phys. Rev. E, 78 (2008) [14] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, vol 19 (1984), Berlin: Springer, Berlin · Zbl 0558.76051 [15] Kuramoto, Y.; Battogtokh, D., Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenom. Complex Syst., 380-385 (2002) [16] Larger, L.; Penkovsky, B.; Maistrenko, Y., Laser chimeras as a paradigm for multistable patterns in complex systems, Nat. Commun., 6, 7752 (2015) [17] Martens, E. A.; Thutupalli, S.; Fourrière, A.; Hallatschek, O., Chimera states in mechanical oscillator networks, Proc. Natl Acad. Sci., 110, 10563-10567 (2013) [18] Marvel, S. A.; Mirollo, R. E.; Strogatz, S. H., Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action, Chaos, 19 (2009) · Zbl 1311.34082 [19] Omel’chenko, O. E., Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Nonlinearity, 26, 2469 (2013) · Zbl 1281.34051 [20] Omel’chenko, O. E., The mathematics behind chimera states, Nonlinearity, 31, R121-R164 (2018) · Zbl 1395.34045 [21] Panaggio, M. J.; Abrams, D. M., Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity, 28, R67-R87 (2015) · Zbl 1392.34036 [22] Sanders, J. A.; Verhulst, F.; Murdock, J., Averaging Methods in Nonlinear Dynamical Systems (2007), New York: Springer, New York · Zbl 1128.34001 [23] Toenjes, R.; Fiore, C.; Pereira, T., Network induced coherence resonance, Nat. Commun., 12 (2021) [24] Totz, J. F.; Rode, J.; Tinsley, M. R.; Showalter, K.; Engel, H., Spiral wave chimera states in large populations of coupled chemical oscillators, Nat. Phys., 14, 282-285 (2018) [25] Vlasov, V.; Pikovsky, A.; Macau, E. E N., Star-type oscillatory networks with generic Kuramoto-type coupling: a model for japanese drums synchrony, Chaos, 25 (2015) · Zbl 1374.34123 [26] Vlasov, V.; Zou, Y.; Pereira, T., Explosive synchronization is discontinuous, Phys. Rev. E, 92 (2015) [27] Watanabe, S.; Strogatz, S. H., Constants of motion for superconducting josephson arrays, Physica D, 74, 197-253 (1994) · Zbl 0812.34043 [28] Wolfrum, M.; Omel’chenko, E., Chimera states are chaotic transients, Phys. Rev. E, 84 (2011) · Zbl 1345.34067 [29] Wolfrum, M.; Omel’chenko, O. E.; Yanchuk, S.; Maistrenko, Y. L., Spectral properties of chimera states, Chaos, 21 (2011) · Zbl 1345.34067
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