#### For citation:

Shabunin A. V. Multistability of periodic orbits in ensembles of maps with long-range couplings. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 2, pp. 5-23. DOI: 10.18500/0869-6632-2018-26-2-5-23

# Multistability of periodic orbits in ensembles of maps with long-range couplings

Aim. The aim of the investigation is to study the regularities of phase multistability in an ensemble of oscillatory systems with non-local couplings in dependance of strength and radius of the couplings, as well as to describe them from the point of view of the spatial spectrum. Method. Study has been carried out by means of numerical simulation of ensemble of logistic maps, by calculation of the phase differences between the oscillations in the subsystems, which define spatial phase clusters and analyze their spectra. The structure of the couplings has been considered as a digital filter whose frequency characteristic depends on the couplings parameters. Results. The research has revealed that the ensemble of maps with period-doubling bifurcations demonstrates developed phase multistability at weak couplings. While the couplings strength and radius grow, the number of coexisting regimes decrease monotonically. Then, if the ensemble is not too large, the multistability is changed by the globally stable regime of in-phase synchronization. The plot of the number of attractors in depending of the coupling strength has the form of steps. At small coupling there is a finite range where the number of the regimes reaches its maximal value and this range practically not depends on the radius of the couplings. Then, while the couplings strength increases, the number of attractors decreases abruptly from step to step. The order of the attractors disappearance is determined by their spatial cluster structure. It can be explained by considering this process as a result of spatial filtering, when the system of the ensemble couplings operates as a digital filter. The wavelength characteristic of the last defines the order of disappearance of clusters. Discussion. The most interesting result is the discovered effect of a jump-like change of the quantity of the attractors with the strength of the couplings. It can be explained if we consider the system of couplings as a spatial filter, the bandwidth of which depends on the couplings parameters. The use of methods of spatial spectra seems to give promising perspectives for the analysis of dynamics of networks with complex topology, including the study of synchronization and multistability in chaotic oscillators and maps. The discovered regularities generalize the results known for ensembles oscillators with local couplings. They are also applicable to ensembles of self-oscillating systems with continuous time, as well as to the phenomenon of multistability in systems with chaotic dynamics and period-doubling bifurcations.

- Ermentrout G.B. J. of Math. Biol., 1985, vol. 22, pp. 1–9.
- Crawford J.D., Davies K.T.R. Physica D., 1990, vol. 125, pp. 1–46.
- Ren L., Ermentrout G.B. Physica D., 2000, vol. 143, p. 56.
- Kuramoto Y. Chemical oscillations, waves and turbulence. Berlin: Springer. 1984.
- Cross M.G., Hohenberg P.C. Rev. Mod. Phys., 1993, vol. 65, no. 3, pp. 851–1112.
- Mosekilde E., Maistrenko Yu., Postnov D. Chaotic Synchronization. Applications to Living Systems. Singapore: World Scientific, 2002.
- Abrams D.M., Strogatz S.H. Phys. Rev. Lett., 2004, vol. 93, p. 174102.
- Omelchenko I., Maistrenko Y., Hovel P., Scholl E. Phys. Rev. Lett., 2011, vol. 106, p. 234102.
- Hagerstrom A.M., Murphy T.E., Roy R., Hoevel P., Omelchenko I., Scholl E. Nature Physics, 2012, vol. 8, pp. 658–661.
- Bogomolov S.A., Strelkova G.I., Anishchenko V.S., Scholl E. Technical Physics Letters, 2016, vol. 42, no. 7, pp. 765–768.
- Gopal R., Chandrasekar V.K., Venkatesan A., Lakshmanan M. Phys. Rev. E., 2014, vol. 89, p. 052914.
- Arecchi F.T., Meucci R., Puccioni G., Tredicce J. Phys. Rev. Lett., 1982, vol. 49, pp. 1217–1220.
- Prengel F., Wacker A., Scholl E. Phys. Rev. B., 1994, vol. 50, pp. 1705–1712.
- Sun N.G., Tsironis G.P. Phys. Rev. B., 1995, vol. 51, pp. 11221–11224.
- Foss J., Longtin A., Mensour B., Milton J. Phys. Rev. Lett., 1996, vol. 76, pp. 708–711.
- Ermentrout G.B. J. of Math. Biol., 1985, vol. 23, no. 1, pp. 55–74.
- Ermentrout G.B. SIAM J. of Appl. Math., 1992, vol. 52, no. 6, pp. 1664–1687.
- Shabunin A.V., Akopov A.A., Astakhov V.V., Vadivasova T.E. Izvestija VUZ, Applied Nonlinear Dynamics, 2005, vol. 13, no. 4, pp. 37–54 (in Russian).
- Astakhov V.V., Bezruchko B.P., Gulyaev Y.P., Seleznev E.P. Technical Physics Letters, 1989, vol. 15, no. 3, pp. 60–65 (in Russian).
- Astakhov V.V., Bezruchko B.P., Pudovochkin O.B., Seleznev E.P. Radiotekhnika i elektronika, 1993, vol. 38, no. 2, pp. 291–295 (in Russian).
- Astakhov V., Shabunin A., Kapitaniak T., Anishchenko V. Phys. Rev. Lett., 1997, vol. 79, p. 1014.
- Astakhov V., Shabunin A., Uhm W., Kim S. Phys. Rev. E., 2001, vol. 63, p. 056212.
- Astakhov V.V., Shabunin A.V., Anishchenko V.S. Journal of Communications Technology and Electronics, 1997, vol. 42, p. 907.
- Bezruchko B.P., Prokhorov M.D., Seleznev E.P. Chaos, Solitons and Fractals, 2003, vol. 15, p. 695–711.

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