Geodesic
Attractors of Singularly Perturbed Parabolic Systems of First Degree of Nonroughness in a Plane Domain
Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 663-669.

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We study the problem of the attractors of the boundary-value problem $$ u_t=\sqrt \varepsilon (D_0 + \sqrt \varepsilon D_1)\Delta u + (A_0 + \varepsilon A_1)u + F(u),\qquad u_x|_{x=0,x=l_1} = u_y|_{y=0,y=l_2}=0, $$ where $0\le\varepsilon\ll 1$, $u\in \mathbb{R}^N$, $N\ge 3$, $\Delta $ is the Laplace operator, and $-D_0$ is the Hurwitz matrix. For such a boundary-value problem, under certain assumptions, we establish the existence of any finite fixed number of stable cycles, provided that $\varepsilon>0$ is chosen appropriately small. In other words, it is shown that this boundary-value problem involves the buffer phenomenon. 
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A. Yu. Kolesov; A. N. Kulikov; N. Kh. Rozov. Attractors of Singularly Perturbed Parabolic Systems of First Degree of Nonroughness in a Plane Domain. Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 663-669. http://geodesic.mathdoc.fr/item/MZM_2004_75_5_a2/

[1] Kolesov Yu. S., “Dinamika biologicheskikh populyatsii”, Mezhvuz. sb., Gorkovskii un-t, 1984, 28–34

[2] Kolesov Yu. S., “Ob odnoi bifurkatsionnoi teoreme v teorii avtokolebanii raspredelennykh sistem”, Differents. uravneniya, 21:10 (1985), 1709–1713 | MR | Zbl

[3] Kolesov Yu. S., “Bifurkatsiya invariantnykh torov parabolicheskikh sistem s maloi diffuziei”, Matem. sb., 184:3 (1993), 121–136 | Zbl

[4] Zakharov A. A., Kolesov Yu. S., Nelineinye kolebaniya i ekologiya, YarGU, Yaroslavl, 1984, 3–15

[5] Kolesov A. Yu., “Bifurkatsiya periodicheskikh reshenii singulyarno vozmuschennykh parabolicheskikh sistem pervoi stepeni negrubosti”, Ukr. matem. zhurn., 42:8 (1990), 1037–1042 | MR

[6] Sobolevskii P. E., “Ob uravneniyakh parabolicheskogo tipa v banakhovom prostranstve”, Tr. MMO, 10, URSS, M., 1961, 297–350 | MR | Zbl

[7] Kolesov A. Yu., Rozov N. Kh., “Osobennosti dinamiki uravneniya Ginzburga–Landau v ploskoi oblasti”, TMF, 125:2 (2000), 205–220 | MR | Zbl

[8] Kolesov Yu. S., Kharkov A. E., “Attraktory singulyarno vozmuschennykh sistem tipa reaktsiya-diffuziya s krainei tochkoi povorota v ploskoi oblasti”, Dokl. RAN, 377:2 (2001), 173–175 | MR