Geodesic
Dynamics of coupled Van der Pol oscillators
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 53-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider the problem on the synchronization of two and three weakly coupled Van der Pol oscillators in the case where the oscillators are identical and the connection between them is weak. The existence and stability of two types of periodic solutions are studied under the assumption that the connection between the oscillators is dissipative or active. The analysis of the problem is based on methods of the qualitative theory of differential equations, namely, the method of integral manifolds and the method of normal Poincaré–Dulac forms. The problem is reduced to the study of normal forms. We used a version of the Krylov–Bogolyubov algorithm, which allows one to obtain asymptotic formulas for periodic solutions.
Keywords: oscillator, weak connection, periodic solution, stability, normal form, synchronization. 
@article{INTO_2019_168_a6,
     author = {D. A. Kulikov},
     title = {Dynamics of coupled {Van} der {Pol} oscillators},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {53--60},
     publisher = {mathdoc},
     volume = {168},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_168_a6/}
}
TY  - JOUR
AU  - D. A. Kulikov
TI  - Dynamics of coupled Van der Pol oscillators
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 53
EP  - 60
VL  - 168
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_168_a6/
LA  - ru
ID  - INTO_2019_168_a6
ER  - 
%0 Journal Article
%A D. A. Kulikov
%T Dynamics of coupled Van der Pol oscillators
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 53-60
%V 168
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_168_a6/
%G ru
%F INTO_2019_168_a6
D. A. Kulikov. Dynamics of coupled Van der Pol oscillators. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 53-60. http://geodesic.mathdoc.fr/item/INTO_2019_168_a6/

[1] Demidovich B. P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR

[2] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa tochechnykh otobrazhenii: printsip koltsa”, Differ. uravn., 39:5 (2003), 584–601 | MR | Zbl

[3] Kolesov A. Yu., Kulikov A. N., Rozov N. Kh., “Invariantnye tory odnogo klassa otobrazhenii: sokhranenie tora pri vozmuscheniyakh”, Differ. uravn., 39:6 (2003), 738–752 | MR

[4] Kuznetsov A. P., Paksyutov V. I., “O dinamike dvukh slabosvyazannykh ostsillyatorov Van der Polya—Duffinga s dissipativnoi svyazyu”, Izv. vuzov. Prikl. nelin. dinam., 11:6 (2003), 48–63

[5] Kuznetsov A. P., Tyuryukina L. V., Sataev I. R., Chernyshov N. Yu., “Sinkhronizatsiya i mnogochastotnye kolebaniya v nizkorazmernykh ansamblyakh ostsillyatorov”, Izv. vuzov. Prikl. nelin. dinam., 22:1 (2014), 27–54 | Zbl

[6] Kulikov D. A., “Avtomodelnye periodicheskie resheniya i bifurkatsii ot nikh v zadache o vzaimodeistvii dvukh slabosvyazannykh ostsillyatorov”, Izv. vuzov. Prikl. nelin. dinam., 14:5 (2006), 120–132 | Zbl

[7] Kulikov D. A., “Tsikly bilokalnoi modeli volnovogo uravneniya: polnyi analiz”, Sovrem. probl. mat. inform., 4 (2001), 93–96

[8] Kulikov D. A., “Issledovanie dinamiki bilokalnoi modeli nelineinykh volnovykh uravnenii”, Sovrem. probl. mat. inform., 5 (2002), 46–52

[9] Kulikov D. A., Attraktory uravneniya Ginzburga—Landau i ego konechnomernogo analoga, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, 2006

[10] Pikovskii A., Rozenblyum M., Kurtts Yu., Sinkhronizatsiya. Fundamentalnoe yavlenie, Tekhnosfera, M., 2003

[11] Kulikov D. A., “Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators”, J. Appl. Math. Mech., 74:4 (2010), 389–400 | DOI | MR | Zbl