Attraction basins of clusters in coupled map lattices

Matvey P. Kulakov ; Efim Ya. Frisman

Russian journal of nonlinear dynamics, Tome 11 (2015) no. 1, pp. 51-76.

Résumé

This paper researches a phenomenon of clustering and multistability in a non-global coupled Ricker maps. To construct attraction basins for some phases of clustering we propose a method. For this purpose we consider the several simultaneously possible and fundamentally different trajectories of the system corresponding to different phases of clustering. As a result these phases or trajectories have the unique domains of attraction (basins) in the phase space and stability region in the parametric space. The suggested approach consists in that each a trajectory is approximated the non-identical asymmetric coupled map lattices consisting of fewer equations and equals the number of clusters. As result it is shown the formation and transformation of clusters is the same like a bifurcations leading to birth of asynchronous modes in approximating systems.

Détail

Mots-clés : metapopulation
Keywords: multistability, coupled map lattices, clustering, basin of attraction.

Bibliographie

[1] Kaneko K., “Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled logistic lattice: Towards a prelude of a «field theory of chaos»”, Progr. Theoret. Phys., 72:3 (1984), 480–486 | DOI

[2] Kuznetsov S. P., “O modelnom opisanii tsepochki svyazannykh dinamicheskikh sistem vblizi tochki perekhoda poryadok–besporyadok”, Izv. vuzov. Fizika, 27:6 (1984), 87–96

[3] Oppo G.-L., Kapral R., “Discrete models for the formation and evolution of spatial structure in dissipative systems”, Phys. Rev. A, 33:6 (1984), 4219–4231 | DOI

[4] Waller I., Kapral R., “Spatial and temporal structure in systems of coupled nonlinear oscillators”, Phys. Rev. A, 30:4 (1986), 2047–2055 | DOI

[5] Kaneko K., “Lyapunov analysis and information flow in coupled map lattices”, Phys. D, 23:1–3 (1986), 436–447 | DOI

[6] Kaneko K., “Clustering, coding, switching, hierarchical, ordering, and control in network of chaotic elements”, Phys. D, 41:2 (1990), 137–172 | DOI

[7] Kaneko K., “Relevance of dynamic clustering to biological networks”, Phys. D, 75:1–3 (1994), 55–73 | DOI

[8] Kuznetsov S. P., “Universalnost i podobie svyazannykh sistem Feigenbauma”, Izv. vuzov. Radiofizika, 27:8 (1985), 991–1007

[9] Kuznetsov A. P., Kuznetsov S. P., “Kriticheskaya dinamika reshetok svyazannykh otobrazhenii u poroga khaosa”, Izv. vuzov. Radiofizika, 34:10–12 (1991), 1079–1115

[10] Manica V., Silva J. A., “Population distribution and synchronized dynamics in a metapopulation model in two geographic scales”, Math. Biosci., 250 (2014), 1–9 | DOI

[11] Martins L. C., Brunnet L. G., “Multi-state coupled map lattices”, Phys. A, 296:1–2 (2001), 119–130 | DOI

[12] Richter H., “Coupled map lattices as spatio-temporal fitness functions: Landscape measures and evolutionary optimization”, Phys. D, 237:2 (2008), 167–186 | DOI

[13] Kulakov M. P., Revutskaya O. L., “Primenenie metapopulyatsionnogo podkhoda k analizu prostranstvenno-vremennoi dinamiki promyslovykh zhivotnykh (na primere populyatsii kabana i izyubrya)”, Regionalnye problemy, 14:2 (2011), 12–20

[14] Ivanova A. S., Kuznetsov S. P., “O dinamike na poroge vozniknoveniya khaosa v modelnykh sistemakh, postroennykh na baze logisticheskikh otobrazhenii s inertsialnoi i dissipativnoi globalnoi svyazyu”, PND, 10:6 (2002), 42–53

[15] Ivanova A. S., Kuznetsov S. P., “Volny klasterizatsii v tsepochke sistem, kazhdaya iz kotorykh soderzhit nabor elementov s vnutrennei globalnoi svyazyu”, PND, 11:4–5 (2003), 80–88

[16] Popovych O., Pikovsky A., Maistrenko Yu., “Cluster-splitting bifurcation in a system of coupled maps”, Phys. D, 168/169 (2002), 106–125 | DOI

[17] Omelchenko I., Maistrenko Yu., Mosekilde E., “Synhronization in ensembles of coupled maps with a major element”, Discrete Dyn. Nat. Soc., 2005:3 (2005), 239–255 | DOI

[18] Chandrasekaran L., Matveev V., Bose A., “Multistability of clustered states in a globally inhibitory network”, Phys. D, 238:3 (2009), 253–263 | DOI

[19] Chen L., Lü J., Lu J., Hill D. J., “Local asymptotic coherence of time-varying discrete ecological networks”, Automatica, 45:2 (2009), 546–552 | DOI

[20] Kulakov M. P., Frisman E. Ya., “Sinkhronizatsiya $2$-tsiklov v sisteme simmetrichno svyazannykh populyatsii, zapas–popolnenie v kotorykh opisyvaetsya funktsiei Rikera”, PND, 18:6 (2010), 25–41

[21] Kulakov M. P., Aksenovich T. I., Frisman E. Ya., “Podkhody k opisaniyu prostranstvennoi dinamiki migratsionno-svyazannykh populyatsii: Analiz sinkhronizatsii tsiklov”, Regionalnye problemy, 16:1 (2013), 5–15

[22] Zhang Y.-Q., Wang X.-Y., “Spatiotemporal chaos in mixed linear–nonlinear coupled logistic map lattice”, Phys. A, 402 (2014), 104–118 | DOI

[23] Opdam P., “Metapopulation theory and habitat fragmentation: A review of holarctic breeding bird studies”, Landscape Ecol., 5:2 (1991), 93–106 | DOI

[24] I. Hanski, O. E. Gaggiotti (eds.), Ecology, genetics and evolution of metapopulations, Acad. Press, London, 2004, 696 pp.

[25] Skaletskaya E. I., Frisman E. Ya., Shapiro A. P., Diskretnye modeli chislennosti populyatsii i optimizatsiya promysla, Nauka, M., 1979, 164 pp.

[26] Feigenbaum M., “Universalnost v povedenii nelineinykh sistem”, UFN, 141:10 (1983), 343–374 | DOI

[27] Yakubu A.-A., Castillo-Chavez C., “Interplay between local dynamics and disperal in discrete-time metapopulation model”, J. Theor. Biol., 218:3 (2002), 273–288 | DOI

[28] Cressman R., Křivan V., “Migration dynamics for the ideal free distribution”, Am. Nat., 168:3 (2006), 384–987 | DOI

[29] Křivan V., Cressman R., Schneider C., “The ideal free distribution: A review and synthesis of the game-theoretic perspective”, Theor. Popul. Biol., 73:3 (2008), 403–425 | DOI

[30] Pisarchik A. N., Feudel U., “Control of multistability”, Phys. Rep., 540:4 (2014), 167–218 | DOI

[31] Bezruchko B. P., Prokhorov M. D., Seleznev E. P., “Vidy kolebanii, multistabilnost i basseiny prityazheniya attraktorov simmetrichno svyazannykh sistem s udvoeniem perioda”, PND, 10:4 (2002), 47–67

[32] Udwadia F. E., Raju N., “Dynamics of coupled nonlinear maps and its application to ecological modeling”, Appl. Math. Comput., 82:2–3 (1997), 137–179 | DOI

[33] Wysham D. B., Hastings A., “Sudden shift ecological systems: Intermittency and transients in the coupled Ricker population model”, Bull. Math. Biol., 70:4 (2008), 1013–1031 | DOI