Selection of spatial modes in an ensemble of non-locally coupled chaotic maps

A. V. Shabunin

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 30 (2022) no. 1, pp. 109-124.

Résumé

Purpose of this work is to determine regularities of formation of spatial structures in an ensemble of chaotic systems with non-local diffusion couplings and to study how these structures depend on the wave response of the digital filter formed by the ensemble couplings structure. Methods. The study was carried out by numerical simulation of an ensemble of logistic maps, calculation of its typical oscillatory regimes and their spectral analysis. The network was considered as a digital filter with a frequency response depending on the coupling parameters. Correlation between the spatial spectra and the amplitude-frequency response of the coupling filter and the mutual coherence of oscillations when the coupling parameters change were considered. Results. The system of couplings between chaotic maps behaves like a wave filter with selective properties, allowing spatial modes with certain wavelengths to exist and suppressing others. The selection of spatial modes is based on the wave characteristic of the coupling filter, the type of which is determined by the radius and the magnitude of couplings. At strong coupling the wave characteristics for ensembles with local and non-local couplings are qualitatively different, therefore they demonstrate essencially different behavior. Discussion. Using spectral methods for dynamics analysis systems with complex network topologies seems to be a promising approach, especially for research of synchronization and multistability in ensembles of chaotic oscillators and maps. The found regularities generalize the results known for ensembles of maps with local couplings. They also can be applied to ensembles of self-sustained oscillators. 

Détail

Keywords: spatiotemporal chaos, ensembles of maps, synchronization, spatial filtering.

Bibliographie

[1] Anischenko V. S., Postnov D. E., Safonova M. A., “Razmernost i fizicheskie svoistva khaoticheskikh attraktorov tsepochki svyazannykh generatorov”, Pisma v ZhTF, 11:24 (1985), 1505–1509

[2] Anischenko V. S., Aranson I. S., Postnov D. E., Rabinovich M. I., “Prostranstvennaya sinkhronizatsiya i razvitie bifurkatsii v tsepochke svyazannykh ostsillyatorov”, Doklady Akademii nauk SSSR, 286:5 (1986), 1120–1124

[3] Fujisaka H., Yamada T., “Stability theory of synchronized motion in coupled-oscillator systems”, Progress of Theoretical Physics, 69:1 (1983), 32–47 | DOI

[4] Yamada T., Fujisaka H., “Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach”, Progress of Theoretical Physics, 70:5, P (1983), 1240–1248 | DOI

[5] Anishchenko V. S., Vadivasova T. E., Postnov D. E., Safonova M. A., “Synchronization of chaos”, International Journal of Bifurcation and Chaos, 2:3 (1992), 633–644 | DOI

[6] Heagy J. F., Carroll T. L., Pecora L. M., “Synchronous chaos in coupled oscillator systems”, Phys. Rev. E, 50:3 (1994), 1874–1884 | DOI

[7] Ren L., Ermentrout B., “Phase locking in chains of multiple-coupled oscillators”, Physica D, 143:1–4 (2000), 56–73 | DOI

[8] Shabunin A. V., Akopov A. A., Astakhov V. V., Vadivasova T. E., “Beguschie volny v diskretnoi angarmonicheskoi avtokolebatelnoi srede”, Izvestiya vuzov. PND, 13:4 (2005), 37–55 | DOI

[9] Kuramoto Y., Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984, 158 pp. | DOI

[10] Cross M. C., Hohenberg P. C., “Pattern formation outside of equilibrium”, Rev. Mod. Phys., 65:3 (1993), 851–1112 | DOI

[11] Mosekilde E., Maistrenko Y., Postnov D., Chaotic Synchronization: Applications to Living Systems, World Scientific, Singapore, 2002, 440 pp. | DOI

[12] Arecchi F. T., Meucci R., Puccioni G., Tredicce J., “Experimental evidence of subharmonic bifurcations, multistability, and turbulence in a Q-switched gas laser”, Phys. Rev. Lett., 49:17 (1982), 1217–1220 | DOI

[13] Astakhov V. V., Bezruchko B. P., Gulyaev Yu. P., Seleznev E. P., “Multistabilnye sostoyaniya dissipativno svyazannykh feigenbaumovskikh sistem”, Pisma v ZhTF, 15:3 (1989), 60–65

[14] Astakhov V. V., Bezruchko B. P., Pudovochkin O. B., Seleznev E. P., “Fazovaya multistabilnost i ustanovlenie kolebanii v nelineinykh sistemakh s udvoeniem perioda”, Radiotekhnika i elektronika, 38:2 (1993), 291–295

[15] Prengel F., Wacker A., Schöll E., “Simple model for multistability and domain formation in semiconductor superlattices”, Phys. Rev. B, 50:3 (1994), 1705–1712 | DOI

[16] Sun N. G., Tsironis G. P., “Multistability of conductance in doped semiconductor superlattices”, Phys. Rev. B, 51:16 (1995), 11221–11224 | DOI

[17] Foss J., Longtin A., Mensour B., Milton J., “Multistability and delayed recurrent loops”, Phys. Rev. Lett., 76:4 (1996), 708–711 | DOI

[18] Abrams D. M., Strogatz S. H., “Chimera states for coupled oscillators”, Phys. Rev. Lett., 93:17 (2004), 174102 | DOI

[19] Omelchenko I., Maistrenko Y., Hövel P., Schöll E., “Loss of coherence in dynamical networks: Spatial chaos and chimera states”, Phys. Rev. Lett., 106:23 (2011), 234102 | DOI

[20] Hagerstrom A. M., Murphy T. E., Roy R., Hövel P., Omelchenko I., Schöll E., “Experimental observation of chimeras in coupled-map lattices”, Nature Physics, 8:9 (2012), 658–661 | DOI

[21] Bogomolov S. A., Strelkova G. I., Schöll E., Anischenko V. S., “Amplitudnye i fazovye khimery v ansamble khaoticheskikh ostsillyatorov”, Pisma v ZhTF, 42:14 (2016), 103–110

[22] Gopal R., Chandrasekar V. K., Venkatesan A., Lakshmanan M., “Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling”, Phys. Rev. E, 89:5 (2014), 052914 | DOI

[23] Shabunin A., Astakhov V., Kurths J., “Quantitative analysis of chaotic synchronization by means of coherence”, Phys. Rev. E, 72:1 (2005), 016218 | DOI

[24] Shabunin A. V., “Multistabilnost periodicheskikh orbit v ansamble otobrazhenii s dalnodeistvuyuschimi svyazyami”, Izvestiya vuzov. PND, 26:2 (2018), 5–23 | DOI

[25] Shabunin A., “Selective properties of diffusive couplings and their influence on spatiotemporal chaos”, Chaos, 31:7 (2021), 073132 | DOI

[26] Kaneko K., “Pattern dynamics in spatiotemporal chaos: Pattern selection, diffusion of defect and pattern competition intermettency”, Physica D, 34:1–2 (1989), 1–41 | DOI