Geodesic
On repelling cycles and chaotic solutions of difference equations with random parameters
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 2, pp. 227-235 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider difference equations with right-hand sides depending at each moment not only on the value at the preceding moment but also on a parameter that takes random values in a given set $\Omega$. For this probabilistic model, we study various dynamic scenarios, which are in a certain way different from scenarios of deterministic models and give a more comprehensive presentation of the processes in real physical systems. We derive conditions for the existence of attracting and repelling cycles of length $k\geqslant 1$ that hold for all values of the random parameter and conditions that hold with probability one. We also derive conditions under which the solutions are chaotic with probability one. It is shown that the chaotic solutions exist in the case where either the equation with random parameters has no cycles or all the cycles are repelling with probability one.
Keywords: difference equations with random parameters, attracting and repelling cycles, chaotic trajectory. 
@article{TIMM_2016_22_2_a24,
     author = {L. I. Rodina},
     title = {On repelling cycles and chaotic solutions of difference equations with random parameters},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {227--235},
     year = {2016},
     volume = {22},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2016_22_2_a24/}
}
TY  - JOUR
AU  - L. I. Rodina
TI  - On repelling cycles and chaotic solutions of difference equations with random parameters
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2016
SP  - 227
EP  - 235
VL  - 22
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2016_22_2_a24/
LA  - ru
ID  - TIMM_2016_22_2_a24
ER  - 
%0 Journal Article
%A L. I. Rodina
%T On repelling cycles and chaotic solutions of difference equations with random parameters
%J Trudy Instituta matematiki i mehaniki
%D 2016
%P 227-235
%V 22
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2016_22_2_a24/
%G ru
%F TIMM_2016_22_2_a24
L. I. Rodina. On repelling cycles and chaotic solutions of difference equations with random parameters. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 2, pp. 227-235. http://geodesic.mathdoc.fr/item/TIMM_2016_22_2_a24/

[1] Svirezhev Yu.M., Logofet D.O., Ustoichivost biologicheskikh soobschestv, Nauka, M., 1978, 352 pp. | MR

[2] A.N. Sharkovskii, S.F. Kolyada, A.G. Sivak, V.V. Fedorenko, Dinamika odnomernykh otobrazhenii, Naukova dumka, Kiev, 1989, 216 pp. | MR

[3] Riznichenko G.Yu., Lektsii po matematicheskim modelyam v biologii. Ch. 1., NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2002, 232 pp.

[4] Sharkovskii A.N., “Sosuschestvovanie tsiklov nepreryvnogo preobrazovaniya pryamoi v sebya”, Ukr. mat. zhurn., 16:1 (1964), 61–71

[5] Li Tien-Yien, Yorke James A., “Period three implies chaos”, Amer. Math. Monthly, 82:10 (1975), 985–992 | DOI | MR | Zbl

[6] Shiryaev A.N., Veroyatnost, Nauka, M., 1989, 580 pp. | MR

[7] Masterkov Yu.V., Rodina L.I., “Dostatochnye usloviya lokalnoi upravlyaemosti sistem so sluchainymi parametrami dlya proizvolnogo chisla sostoyanii sistemy”, Izv. vuzov. Matematika, 2008, no. 3, 38–49 | MR | Zbl

[8] Rodina L.I., “O nekotorykh veroyatnostnykh modelyakh dinamiki rosta populyatsii”, Vestn. Udmurt. un-ta. Matematika. Mekhanika. Kompyut. nauki, 2013, no. 4, 109–124 | Zbl

[9] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, 2, Mir, M., 1984, 738 pp. | MR

[10] Bratus A.S., Novozhilov A.S., Rodina E.V., Diskretnye dinamicheskie sistemy i matematicheskie modeli v ekologii, ucheb. posobie, MIIT, M., 2005, 139 pp.

[11] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, 1, Mir, M., 1984, 529 pp. | MR