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

For citation:

Rybalova E. V., Anishchenko V. S. Influence of noise on spiral and target wave regimes in two-dimensional lattice of locally coupled maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 2, pp. 272-287. DOI: 10.18500/0869-6632-2021-29-2-272-287

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: 1076)
Article type: 

Influence of noise on spiral and target wave regimes in two-dimensional lattice of locally coupled maps

Rybalova E. V., Saratov State University
Anishchenko Vadim Semenovich, Saratov State University

 Abstract. The objective is to study numerically the dynamics of two-dimensional lattice of locally coupled maps of Rulkov. We analyze conditions for the appearance and existence as well as the properties of auto-wave spatio-temporal structures which are represented by spiral and target waves. The influence of noise on the lattice dynamics is explored as the noise intensity and the size of the noise-disturbed region are varied. Methods. In numerical experiments the evolution of the lattice dynamics is directly determined by the corresponding recurrence relations. The numerical data are used to construct spatial distributions of the instantaneous values of the amplitudes for all the network elements, spatio-temporal diagrams for the lattice cross-section at different values of the control parameters of the individual nodes, for various noise intensities and different sizes of the noise-disturbed region. The obtained results are compared. The noise-disturbed region is specified as a square which consists of a small number of oscillators at the lattice center. Results. It is found that for certain values of the control parameters of the maps, of the coupling parameters, and the initial conditions, long-lived spiral and target waves can exist in the lattice. It is shown that the spiral wave regimes are, as a rule, transient, can be observed for a finite time and become long-lived only for certain values of the parameters and the initial conditions. When the noise influences a finite region of the lattice showing spiral waves, the transition to spiral waves with a different structure or to target waves can occur. However, if the noise disturbance is removed, the lattice returns to its original mode or exhibits the transition to coherent dynamics modes. The target waves are more resistant to the noise and are observed for longer times. If the noise causes the target waves to change, the resulting regime continues to exist after removing the noise source. Conclusion. It is shown that the spiral and target waves can be observed in the lattice of locally coupled Rulkov maps. The regions where these waves exist are defined and constructed in the plane of the control parameters of the individual elements. Studying the impact of the relation between the noise intensity and the size of the noise-disturbed region enables one to distinguish the region where the transition from spiral to target waves always occurs, as well as the area inside which this transition depends on the initial states of the lattice elements and the noise realization. The effect of noise on the target waves can induce the appearance of only target wave chimeras which continue to exist even after the noise excitation is turned off.

The reported study was funded by the Russian Science Foundation (project no. 20-12-00119).
  1. Kuramoto Y. Chemical Oscillations, Waves and Turbulence. Springer-Verlag, Berlin, Heidelberg; 1984. DOI: 10.1007/978-3-642-69689-3.
  2. Kaneko K. Pattern dynamics in spatiotemporal chaos: Pattern selection, diffusion of defect and pattern competition intermettency. Physica D: Nonlinear Phenomena. 1989;34(1–2):1–41. DOI: 10.1016/0167-2789(89)90227-3.
  3. Kuznetsov AP, Kuznetsov SP. Critical dynamics of coupled map lattices at the onset of chaos. Radiophysics and Quantum Electronics. 1991;34(10–23):1079–1115. DOI: 10.1007/BF01083617.
  4. Wang DL. Modeling global synchrony in the visual cortex by locally coupled neural oscillators. Computation in Neurons and Neural Systems. Springer, Boston, MA; 1994. P. 109–114. DOI: 10.1007/978-1-4615-2714-5_18.
  5. Nicolis G. Introduction to Nonlinear Science. Cambridge University Press; 1995. DOI: 10.1017/CBO9781139170802.
  6. Mikhailov AS, Loskutov AY. Foundation of Synergetics: Complex Patterns. Vol. 52 of Springer Series in Synergetics. Springer-Verlag, Berlin, Heidelberg; 1991. P. 210. DOI: 10.1007/978-3-642-97294-2.
  7. Afraimovich VS, Nekorkin VI, Osipov GV, Shalfeev VD. Stability, Structures and Chaos in Nonlinear Synchronization Networks. World Scientific Series on Nonlinear Science Series A: Vol. 6. Singapore; 1995. P. 260. DOI: 10.1142/2412.
  8. Belykh VN, Belykh IV, Hasler M. Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems. Physical Review E. 2000;62(5):6332–6345. DOI: 10.1103/physreve.62.6332.
  9. Strogatz SH. Exploring complex networks. Nature. 2001;410(6825):268–276. DOI: 10.1038/35065725.
  10. Dorogovtsev SN, Mendes JF. Evolution of networks. Adv. Phys. 2002;51(4):1079–1187. DOI: 10.1080/00018730110112519.
  11. Newman MEJ. The structure and function of complex networks. SIAM Review. 2003;45(2): 167–256. DOI: 10.1137/S003614450342480.
  12. Ben-Naim E, Frauenfelder H, Toroczkai Z. Complex Networks. Vol. 650 of Lecture Notes in Physics. Springer Science & Business Media; 2004. P. 520. DOI: 10.1007/b98716.
  13. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU. Complex networks: Structure and dynamics. Physics reports. 2006;424(4–5):175–308. DOI: 10.1016/j.physrep.2005.10.009.
  14. Kuznetsov SP, Pikovsky AS. Autonomous coupled oscillators with hyperbolic strange attractors. Physica D: Nonlinear Phenomena. 2007;232(2):87–102. DOI: 10.1016/j.physd.2007.05.008.
  15. Osipov GV, Kurths J, Zhou C. Synchronization in Oscillatory Networks. Springer Science & Business Media; 2007. DOI: 10.1007/978-3-540-71269-5.
  16. Kuznetsov AP, Kuznetsov SP, Turukina LV, Sataev IR. Landau-Hopf scenario in the ensemble of interacting oscillators. Russian Journal of Nonlinear Dynamics. 2012;8(5):863–873 (in Russian). DOI: 10.20537/nd1205001.
  17. Nekorkin V, Velarde MG. Synergetic Phenomena in Active Lattices: Patterns, Waves, Solitons, Chaos. Springer Science & Business Media; 2002. P. 359. DOI: 10.1007/978-3-642-56053-8.
  18. Castelpoggi F, Wio HS. Stochastic resonant media: Effect of local and nonlocal coupling in reaction-diffusion models. Physical Review E. 1998;57(5):5112–5121. DOI: 10.1103/PhysRevE.57.5112.
  19. Kuramoto Y, Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenomena in Complex Systems. 2002;5(4):380–385.
  20. Tanaka D, Kuramoto Y. Complex Ginzburg-Landau equation with nonlocal coupling. Physical Review E. 2003;68(2):026219. DOI: 10.1103/PhysRevE.68.026219.
  21. Abrams DM, Strogatz SH. Chimera states for coupled oscillators. Physical Review Letters. 2004;93(17):174102. DOI: 10.1103/PhysRevLett.93.174102.
  22. Wolfrum M, Omel’chenko OE, Yanchuk S, Maistrenko YL. Spectral properties of chimera states. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2011;21(1):013112. DOI: 10.1063/1.3563579.
  23. Omel’chenko OE, Wolfrum M, Yanchuk S, Maistrenko YL, Sudakov O. Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators. Physical Review E. 2012;85(3):036210. DOI: 10.1103/PhysRevE.85.036210.
  24. Nkomo S, Tinsley MR, Showalter K. Chimera states in populations of nonlocally coupled chemical oscillators. Physical Review Letters. 2013;110(24):244102. DOI: 10.1103/PhysRevLett.110.244102.
  25. 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(2):25–36. DOI: 10.1016/j.cnsns.2016.06.024.
  26. Kuramoto Y, Shima S. Rotating spirals without phase singularity in reaction-diffusion systems. Progress of Theoretical Physics Supplement. 2003;150:115–125. DOI: 10.1143/PTPS.150.115.
  27. Rybalova EV, Bukh AV, Strelkova G, Anishchenko VS. Spiral and target wave chimeras in a 2D lattice of map-based neuron models. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019;29(10):101104. DOI: 10.1063/1.5126178.
  28. Buh AV, Rybalova EV, Anishchenko VS. Autowave structures in two-dimensional lattices of nonlocally coupled oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(3):299–323 (in Russian). DOI: 10.18500/0869-6632-2020-28-3-299-323.
  29. Rulkov NF. Modeling of spiking-bursting neural behavior using two-dimensional map. Physical Review E. 2002;65(4):041922. DOI: 10.1103/PhysRevE.65.041922.