ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Nechaev V. A., Rybalova E. V., Strelkova G. I. Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 6, pp. 943-952. DOI: 10.18500/0869-6632-2021-29-6-943-952

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 207)
Article type: 

Influence of parameters inhomogeneity on the existence of chimera states in a ring of nonlocally coupled maps

Nechaev Vasiliy Andreevich, Saratov State University
Rybalova E. V., Saratov State University
Strelkova Galina Ivanovna, Saratov State University

The aim of the research is to study the influence of inhomogeneity in a control parameter of all partial elements in a ring of nonlocally coupled chaotic maps on the possibility of observing chimera states in the system and to compare the changes in regions of chimera realization using different methods of introducing the inhomogeneity. Methods. In this paper, snapshots of the system dynamics are constructed for various values of the parameters, as well as spatial distributions of cross-correlation coefficient values, which enable us to determine the regime observed in the system for these parameters. To improve the accuracy of the obtained results, the numerical studies are carried out for fifty different realizations of initial conditions of the ring elements. Results. It is shown that a fixed inhomogeneous distribution of the control parameters with increasing noise intensity leads to an increase in the range of the coupling strength where chimera states are observed. With this, the boundary lying in the region of strong coupling changes more significantly as compared with the case of weak coupling strength. The opposite effect is provided when the control parameters are permanently affected by noise. In this case increasing the noise intensity leads to a decrease in the interval of existence of chimera states. Additionally, the nature of the random variable distribution (normal or uniform one) does not strongly influence the observed changes in the ring dynamics. The regions of existence of chimera states are constructed in the plane of «coupling strength – noise intensity» parameters. Conclusion. We have studied how the region of existence of chimeras changes when the coupling strength between the ring elements is varied and when different characteristics of the inhomogeneous distribution of the control parameters are used. It has been shown that in order to increase the region of observing chimera states, the control parameters of the elements must be distributed inhomogeneously over the entire ensemble. To reduce this region, a constant noise effect on the control parameters should be used.

The research was carried out in the framework of the grant of the Russian Science Foundation (project no. 20-12-00119)
  1. Nekorkin V, Velarde MG. Synergetic Phenomena in Active Lattices: Patterns, Waves, Solitons, Chaos. Berlin Heidelberg: Springer-Verlag; 2002. 359 p. DOI: 10.1007/978-3-642-56053-8.
  2. Barrat A, Barthelemy M, Vespignani A. Dynamical Processes on Complex Networks. Cambridge: Cambridge University Press; 2008. 347 p. DOI: 10.1017/CBO9780511791383.
  3. Boccaletti S, Pisarchik AN, del Genio CI, Amann A. Synchronization: From Coupled Systems to Complex Networks. Cambridge: Cambridge University Press; 2018. 255 p. DOI: 10.1017/9781107297111.
  4. Kuramoto Y, Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems. 2002;5(4):380–385.
  5. Abrams DM, Strogatz SH. Chimera states for coupled oscillators. Phys. Rev. Lett. 2004;93(17): 174102. DOI: 10.1103/PhysRevLett.93.174102.
  6. Omelchenko I, Maistrenko Y, Hovel P, Scholl E. Loss of coherence in dynamical networks: Spatial chaos and chimera states. Phys. Rev. Lett. 2011;106(23):234102. DOI: 10.1103/PhysRevLett.106.234102.
  7. Panaggio MJ, Abrams DM. Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity. 2015;28(3):R67. DOI: 10.1088/0951-7715/28/3/R67.
  8. Rybalova EV, Strelkova GI, Anishchenko VS. Mechanism of realizing a solitary state chimera in a ring of nonlocally coupled chaotic maps. Chaos, Solitons & Fractals. 2018;115:300–305. DOI: 10.1016/j.chaos.2018.09.003.
  9. Semenova N, Zakharova A, Anishchenko V, Scholl E. Coherence-resonance chimeras in a network of excitable elements. Phys. Rev. Lett. 2016;117(1):014102. DOI: 10.1103/PhysRevLett.117.014102.
  10. Vadivasova TE, Slepnev AV, Zakharova A. Control of inter-layer synchronization by multiplexing noise. Chaos. 2020;30(9):091101. DOI: 10.1063/5.0023071.
  11. Loos SAM, Claussen JC, Scholl E, Zakharova A. Chimera patterns under the impact of noise. Phys. Rev. E. 2016;93(1):012209. DOI: 10.1103/PhysRevE.93.012209.
  12. Semenova NI, Strelkova GI, Anishchenko VS, Zakharova A. Temporal intermittency and the lifetime of chimera states in ensembles of nonlocally coupled chaotic oscillators. Chaos. 2017; 27(6):061102. DOI: 10.1063/1.4985143. 
  13. Rybalova EV, Klyushina DY, Anishchenko VS, Strelkova GI. Impact of noise on the amplitude chimera lifetime in an ensemble of nonlocally coupled chaotic maps. Regular and Chaotic Dynamics. 2019;24(4):432–445. DOI: 10.1134/S1560354719040051.
  14. Bogomolov SA, Slepnev AV, Strelkova GI, Scholl E, Anishchenko VS. Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems. Communications in Nonlinear Science and Numerical Simulation. 2017;43:25–36. DOI: 10.1016/j.cnsns.2016.06.024.
  15. Malchow AK, Omelchenko I, Scholl E, Hovel P. Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps. Phys. Rev. E. 2018;98(1):012217. DOI: 10.1103/PhysRevE.98.012217.