BSU Bulletin. Mathematics, Informatics
Bibliographic description:
HOMOTOPY GROUPS OF CONNECTED COMPONENTS OF THE SET OF ROUGH HOMOGENEOUS POLYNOMIAL DIFFERENTIAL EQUATIONS ON THE CIRCLE // BSU Bulletin. Mathematics, Informatics. - 2020. №3. . - С. 3-11.
Title:
HOMOTOPY GROUPS OF CONNECTED COMPONENTS OF THE SET OF ROUGH HOMOGENEOUS POLYNOMIAL DIFFERENTIAL EQUATIONS ON THE CIRCLE
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The article considers differential equations on the circle, the right-hand sides of which are homogeneous trigonometric polynomials of degree n. The set E of such equations is identified with the numerical space of ordered sets of coefficients of the corresponding trigonometric polynomials. An equation from E is called rough if the topological structure of its phase portrait does not change when passing to a close equation. The set of rough equations is open and everywhere dense in the space E. We have described the connected components of the set of rough equations and computed their homotopy groups. For connected components containing equations with singular points, the fundamental group is isomorphic to the group Z of integer, and the remaining homotopy groups are zero. For even n, there are also two connected components consisting of equations without singular points. These components are contractible
Keywords:
differential equation on the circle; trigonometric polynomial; roughness; connected component; homotopy group.
List of references:
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2. Roitenberg V. Sh. O svyaznykh komponentakh mnozhestva vektornykh polei Morsa–Smeila na dvumernykh mnogoobraziyakh [On the Connected Components of the Set of Morse–Smale Vector Fields on Two-Dimensional Manifolds]. Trudy vtorykh Kolmogorovskikh chtenii — Proc. 2nd Kolmogorov Readings. Yaroslavl, 2004. Pp. 352–358.
3. Nozdrinova E. V. Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle. Russian Journal of Nonlinear Dynamics. 2018. Vol. 14, no. 4. Pp. 543–551. DOI: 10.20537/nd180408
4. Roitenberg V. Sh. Grubost vektornykh polei na ploskosti, invariantnykh otnositelno gruppy vrashchenii [Structural Stability of Planar Vector Fields that are Invariant under the Rotation Group]. Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Ser. Matematika. Fizika. 2018. Vol. 50, no. 4. Pp. 398–404. DOI: 10.18413/2075-4639-2018-50-4-398-404
5. Roitenberg V. Sh. O strukture prostranstva odnorodnykh polinomialnykh differentsialnykh uravnenii na okruzhnosti [The Structure of the Space of Polynomial Differential Equations on the Circle]. Vestnik Yuzhno-Uralskogo gosudarstvennogo universiteta. Ser. Matematika. Mekhanika. Fizika. 2020. Vol. 12, no. 2. Pp. 21–30.DOI: 10.14529/mmph200203
6. Rokhlin V. A., Fuks D. B. Nachalnyi kurs topologii. Geometricheskie glavy [Beginner Course of Topology. Geometric Chapters]. Moscow: Nauka Publ., 1977. 488 p.
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