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

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Buh A. V., Rybalova E. V., Anishchenko V. S. Autowave structures in two-dimensional lattices of nonlocally coupled oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 3, pp. 299-323. DOI: 10.18500/0869-6632-2020-28-3-299-323

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Autowave structures in two-dimensional lattices of nonlocally coupled oscillators

Buh Andrej Vladimirovich, Saratov State University
Rybalova E. V., Saratov State University
Anishchenko Vadim Semenovich, Saratov State University

Objective. The aim of the research was to compare the dynamics of spiral and target structures including the dynamics of chimera states in ensembles with different nodes. Numeric simulations for autowave structures in two-dimensional ensembles of coupled van der Pol oscillators and Rulkov’s maps were performed. Cases of local and nonlocal coupling between ensemble nodes were considered. Methods. The evolution dynamics of Rulkov’s map lattice is strictly defined with corresponding recurrent formulae. The ensemble of van der Pol oscillators was integrated using the Heun’s method. Snapshots of amplitude values, spatio-temporal diagrams for corresponding sections, phase portraits and time series for single elements were constructed. Moreover, mean values of frequencies of all nodes, and dependencies of instantaneous frequencies on time for selected maps and oscillators. Results were compared. Results. It is shown that classical spiral and target wave regimes were realized at local coupling. More complex structures including chimera states were obtained when a nonlocal coupling was included. Spiral wave chimera structures with single and several incoherent cores were described. A new chimera structure based on target waves (target wave chimera) was shown to be possible. The analysis of features of incoherent cores for both spiral wave chimera and target wave chimera were presented. The results of coupling parameter effect and external noise influence on the dynamics of the ensembles were discussed. Conclusion. Mean values of frequencies were almost the same for all the elements in both ensembles at target wave chimera regimes. The mean values in the incoherent core differed from the ones in coherent area at spiral wave chimera regimes. A transition from target waves to spiral waves was possible when a noise with sufficiently large intensity was introduced. Noise-induced transition from spiral waves to target waves took place in the ensemble of Rulkov’s maps.


1. Winfree A.T. Rotating chemical reaction // Scientific American. 1974. Vol. 230, no. 6. P. 82–95.

2. Vasiliev V.A., Romanovskii Yu.M., Chernavskii D.S., Yakhno V.G. Autowave Processes in Kinetic Systems. Springer, Netherlands, 1987.

3. Kopell N., Howard L.N. Plane wave solutions to reaction–diffusion equations // Studies in Applied Mathematics. 1973. Vol. 52, no. 4. P. 291–328.

4. Howard L.N., Kopell N. Slowly varying waves and shock structures in reaction–diffusion equations // Studies in Applied Mathematics. 1977. Vol. 56, no. 2. P. 95–145.

5. Hagan P.S. Spiral waves in reaction-diffusion equations // SIAM J. Appl. Math. 1982. Vol. 42, no. 4. P. 762–786.

6. Keener J.P., Tyson J.J. Spiral waves in the Belousov–Zhabotinskii reaction // Physica D: Nonlinear Phenomena. 1986. Vol. 21, no. 2. P. 307–324.

7. Tyson J.J., Keener J.P. Spiral waves in a model of myocardium // Physica D: Nonlinear Phenomena. 1987. Vol. 29, no. 1. P. 215–222.

8. Biktashev V.N. Diffusion of autowaves: Evolution equation for slowly varying autowaves // Physica D: Nonlinear Phenomena. 1989. Vol. 40, no. 1. P. 83–90.

9. Barkley D., Kness M., Tuckerman L.S. Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation // Phys. Rev. A. 1990. Vol. 42, no. 4. P. 2489–2492.

10. Castelpoggi F., Wio H.S. Stochastic resonant media: Effect of local and nonlocal coupling in reaction-diffusion models // Phys. Rev. E. 1998. Vol. 57, no. 5. P. 5112–5121.

11. Ermakova E.A., Panteleev M.A., Shnol E.E. Blood coagulation and propagation of autowaves in flow // Pathophysiology of Haemostasis and Thrombosis. 2005. Vol. 34, no. 2–3. P. 135–142.

12. Krinsky V.I. Autowaves: Results, Problems, Outlooks. Springer Series in Synergetics. Springer, Berlin, Heidelberg, 1984.

13. Ivanitski˘ı G.R., Medvinski˘ı A.B., Tsyganov M.A. From disorder to order as applied to the movement of micro-organisms. Soviet Physics Uspekhi, 1991, vol. 34, no. 4, p. 289. 

14. Zaikin A.N., Zhabotinsky A.M. Concentration wave propagation in two-dimensional liquid-phase self-oscillating system // Nature. 1970. Vol. 225, no. 5232. P. 535–537.

15. Zhabotinsky A., Zaikin A. Autowave processes in a distributed chemical system // Journal of Theoretical Biology. 1973. Vol. 40, no. 1. P. 45–61. 

16. Vasil’ev V.A., Romanovski˘ı Y.M., Yakhno V.G. Autowave processes in distributed kinetic systems. Soviet Physics Uspekhi, 1979, vol. 22, no. 8, p. 615.

17. Krinsky V., Biktashev V., Efimov I. Autowave principles for parallel image processing // Physica D: Nonlinear Phenomena. 1991. Vol. 49, no. 1. P. 247–253.

18. Davidenko J.M., Pertsov A.V., Salomonsz R., Baxter W., Jalife J. Stationary and drifting spiral waves of excitation in isolated cardiac muscle // Nature. 1992. Vol. 355, no. 6358. P. 349–351.

19. Ivanitskii G.R., Medvinskii A.B., Tsyganov M.A. From the dynamics of population autowaves generated by living cells to neuroinformatics. Physics-Uspekhi, 1994, vol. 37, no. 10, p. 961.

20. Winfree A.T. Electrical turbulence in three-dimensional heart muscle // Science. 1994. Vol. 266, no. 5187. P. 1003–1006.

21. Davydov V.A., Morozov V.G., Davydov N.V. Ring-shaped autowaves on curved surfaces // Physics Letters A. 2000. Vol. 267, no. 5–6. P. 326–330.

22. Fenton F.H., Cherry E.M., Hastings H.M., Evans S.J. Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2002. Vol. 12, no. 3. P. 852–892.

23. Hildebrand M., Cui J., Mihaliuk E., Wang J., Showalter K. Synchronization of spatiotemporal patterns in locally coupled excitable media // Physical Review E. 2003. Vol. 68, no. 2. 026205.

24. Kuramoto Y., Shima S. Rotating spirals without phase singularity in reaction-diffusion systems // Progress of Theoretical Physics Supplement. 2003. Vol. 150. P. 115–125.

25. Zhang H., Ruan X.S., Hu B., Ouyang Q. Spiral breakup due to mechanical deformation in excitable media // Physical Review E. 2004. Vol. 70, no. 1. 016212.

26. Shima S., Kuramoto Y. Rotating spiral waves with phase-randomized core in nonlocally coupled oscillators // Physical Review E. 2004. Vol. 69, no. 3. 036213.

27. Kuramoto Y., Shima S.I., Battogtokh D., Shiogai Y. Mean-field theory revives in self-oscillatory fields with non-local coupling // Progress of Theoretical Physics Supplement. 2006. Vol. 161. P. 127–143.

28. Shang L., Yi Z., Ji L. Binary image thinning using autowaves generated by PCNN // Neural Processing Letters. 2007. Vol. 25, no. 1. P. 49–62.

29. Laing C.R. The dynamics of chimera states in heterogeneous Kuramoto networks // Physica D: Nonlinear Phenomena. 2009. Vol. 238, no. 16. P. 1569–1588.

30. Martens E.A., Laing C.R., Strogatz S.H. Solvable model of spiral wave chimeras // Physical Review Letters. 2010. Vol. 104, no. 4. 044101.

31. Kuz’min Y.O. Deformation autowaves in fault zones. Izvestiya, Physics of the Solid Earth, 2012, vol. 48, no. 1, pp. 1–16

32. Nkomo S., Tinsley M.R., Showalter K. Chimera states in populations of nonlocally coupled chemical oscillators // Physical Review Letters. 2013. Vol. 110, no. 24. 244102.

33. Gu C., St-Yves G., Davidsen J. Spiral wave chimeras in complex oscillatory and chaotic systems // Physical Review Letters. 2013. Vol. 111, no. 13. 134101.

34. Tang X., Yang T., Epstein I.R., Liu Y., Zhao Y., Gao Q. Novel type of chimera spiral waves arising from decoupling of a diffusible component // The Journal of Chemical Physics. 2014. Vol. 141, no. 2. 024110.

35. Panaggio M.J., Abrams D.M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators // Nonlinearity. 2015. Vol. 28, no. 3. R67.

36. Xie J., Knobloch E., Kao H.C. Twisted chimera states and multicore spiral chimera states on a two-dimensional torus // Physical Review E. 2015. Vol. 92, no. 4. 042921.

37. Li B.W., Dierckx H. Spiral wave chimeras in locally coupled oscillator systems // Physical Review E. 2016. Vol. 93, no. 2. 020202.

38. Weiss S., Deegan R.D. Weakly and strongly coupled Belousov-Zhabotinsky patterns // Physical Review E. 2017. Vol. 95, no. 2. 022215.

39. Totz J.F., Rode J., Tinsley M.R., Showalter K., Engel H. Spiral wave chimera states in large populations of coupled chemical oscillators // Nature Physics. 2018. Vol. 14, no. 3. P. 282.

40. Kundu S., Majhi S., Muruganandam P., Ghosh D. Diffusion induced spiral wave chimeras in ecological system // The European Physical Journal Special Topics. 2018. Vol. 227, no. 7–9. P. 983–993.

41. Guo S., Dai Q., Cheng H., Li H., Xie F., Yang J. Spiral wave chimera in two-dimensional nonlocally coupled Fitzhugh–Nagumo systems // Chaos, Solitons & Fractals. 2018. Vol. 114. P. 394–399.

42. Bukh A., Strelkova G., Anishchenko V. Spiral wave patterns in a two-dimensional lattice of nonlocally coupled maps modeling neural activity // Chaos, Solitons & Fractals. 2019. Vol. 120. P. 75–82.

43. Omel’chenko O.E., Wolfrum M., Yanchuk S., Maistrenko Y.L., Sudakov O. Stationary patterns of coherence and incoherence in two-dimensional arrays of non-locally-coupled phase oscillators // Physical Review E. 2012. Vol. 85, no. 3. 036210.

44. Tanaka D., Kuramoto Y. Complex Ginzburg-Landau equation with nonlocal coupling // Physical Review E. 2003. Vol. 68, no. 2. 026219.

45. Kuramoto Y., Battogtokh D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators // Nonlinear Phenomena in Complex Systems. 2002. Vol. 5, no. 4. P. 380–385.

46. Abrams D.M., Strogatz S.H. Chimera states for coupled oscillators // Physical Review Letters. 2004. Vol. 93, no. 17. 174102.

47. Wolfrum M., Omel’chenko O.E., Yanchuk S., Maistrenko Y.L. Spectral properties of chimera states // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2011. Vol. 21, no. 1. 013112.

48. Bogomolov S.A., Slepnev A.V., Strelkova G.I., Scholl E., Anishchenko V.S. ¨ Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems // Communications in Nonlinear Science and Numerical Simulation. 2017. Vol. 43. P. 25–36.

49. Bukh A.V., Anishchenko V.S. Spiral, target, and chimera wave structures in a two-dimensional ensemble of nonlocally coupled van der Pol oscillators // Technical Physics Letters. 2019. Vol. 45, no. 7. P. 675–678.

50. Rulkov N.F. Modeling of spiking-bursting neural behavior using two-dimensional map // Physical Review E. 2002. Vol. 65, no. 4. 041922.

51. Panaggio M.J., Abrams D.M. Chimera states on a flat torus // Physical review letters. 2013. Vol. 110, no. 9. 094102.

52. Panaggio M.J. Spot and spiral chimera states: Dynamical patterns in networks of coupled oscillators, Ph.D. thesis, Northwestern University, 2014.

53. Maistrenko Y., Sudakov O., Osiv O., Maistrenko V. Chimera states in three dimensions // New Journal of Physics. 2015. Vol 17, no. 7. 073037.

54. Schmidt A., Kasimatis T., Hizanidis J., Provata A., Hovel P. ¨ Chimera patterns in two-dimensional networks of coupled neurons // Physical Review E. 2017. Vol. 95, no. 3. 032224.

55. Rybalova E., Bukh A., Strelkova G., Anishchenko V. Spiral and target wave chimeras in a 2D lattice of map-based neuron models // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019. Vol. 29, no. 10. 101104.

56. Winfree A.T. Spiral waves of chemical activity // Science. 1972. Vol. 175, no. 4022. P. 634–636. 

57. Fenton F.H., Cherry E.M., Glass L. Cardiac arrhythmia // Scholarpedia. 2008. Vol. 3, no. 7. P. 1665.

58. Zhang H., Hu B., Hu G. Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media // Physical Review E. 2003. Vol. 68, no. 2. 026134.

59. Yuan G., Wang G., Chen S. Control of spiral waves and spatiotemporal chaos by periodic perturbation near the boundary // Europhysics Letters. 2005. Vol. 72, no. 6. P. 908.

60. Yu L., Zhang G., Ma J., Chen Y. Control of spiral waves and spatiotemporal chaos with periodical subthreshold ordered wave perturbations // International Journal of Modern Physics C. 2009. Vol. 20, no. 1. P. 85–96.