Geodesic
Dissipativity in the plane
Sbornik. Mathematics, Tome 75 (1993) no. 1, pp. 125-135 Cet article a éte moissonné depuis la source Math-Net.Ru

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A classical method is considered for studying dissipativity of systems of ordinary differential equations on the plane, consisting in the construction of a system of compact sets covering the plane, whose boundaries are given by trajectories of a certain auxiliary system, and the trajectories of the given system intersect them from outside in (Theorem 1). In this connection the problem of coincidence (Theorem 3) and intersection (Theorems 4–6) of trajectories of two differential inclusions is solved. In conclusion, this method is used to prove Theorem 8, which generalizes certain known results, in particular, theorems by Dragilev, Opial, Reissig, Filippov, and others, on the existence of a periodic solution and the dissipativity of the Liénard and Rayleigh equations, as well as a result of Cartwright and Swinnerton-Dyer, close to Theorem 8. 
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P. N. Savel'ev. Dissipativity in the plane. Sbornik. Mathematics, Tome 75 (1993) no. 1, pp. 125-135. http://geodesic.mathdoc.fr/item/SM_1993_75_1_a6/

[1] Reissig R., Sansone G., Konti R., Kachestvennaya teoriya nelineinykh differentsialnykh uravnenii, Nauka, M., 1974 | MR

[2] Pliss V. A., Nelokalnye problemy teorii kolebanii, Nauka, M., 1964 | MR

[3] Filippov V. V., “Aksiomaticheskaya teoriya prostranstv reshenii obyknovennykh differentsialnykh uravnenii i differentsialnykh vklyuchenii”, DAN SSSR, 280:2 (1985), 304–308 | MR | Zbl

[4] Filippov V. V., “Ob obyknovennykh differentsialnykh uravneniyakh s osobennostyami v pravoi chasti”, Matem. zametki, 38:6 (1985), 832–851 | MR | Zbl

[5] Fedorchuk V. V., Filippov V. V., Obschaya topologiya. Osnovnye konstruktsii, Izd-vo MGU, M., 1988 | Zbl

[6] Aleksandrov P. S., Kombinatornaya topologiya, M.–L., 1947 | MR

[7] Kuratovskii K., Topologiya, t. 2, Mir, M., 1969 | MR

[8] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[9] Savelev P. N., “Dissipativnost sistem obyknovennykh differentsialnykh uravnenii na ploskosti”, Vestn. MGU. Ser. matematika, mekhanika, 1988, no. 4, 93–96 | MR

[10] Filippov A. F., “Differentsialnye uravneniya s razryvnoi pravoi chastyu”, Matem. sbornik, 51:1 (1960), 99–128 | Zbl

[11] Kartrait M. L., Svinnerton-Daier Kh. P. F., “Teoremy ogranichennosti dlya nekotorykh differentsialnykh uravnenii vtorogo poryadka”, Differents. uravneniya, 14:11 (1978), 1941–1979 | MR