On $\omega$-limit sets of nonautonomous differential equations
Klebanov, Boris S.
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994), p. 267-281 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

In this paper the $\omega$-limit behaviour of trajectories of solutions of ordinary differential equations is studied by methods of an axiomatic theory of solution spaces. We prove, under very general assumptions, semi-invariance of $\omega$-limit sets and a Poincar'{e}-Bendixon type theorem.

Publié le : 1994-01-01
Classification:  34A34,  34C05,  34C11,  34C99,  34D05 
@article{118666,
     author = {Boris S. Klebanov},
     title = {On $\omega$-limit sets of nonautonomous differential equations},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {35},
     year = {1994},
     pages = {267-281},
     zbl = {0809.34042},
     mrnumber = {1286574},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118666}
}

Klebanov, Boris S. On $\omega$-limit sets of nonautonomous differential equations. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 267-281. http://gdmltest.u-ga.fr/item/118666/

Artstein Z. Limiting equations and stability of nonautonomous ordinary differential equations, Appendix A in: J.P. LaSalle, {The stability of dynamical systems}, CBMS Regional Conference Series in Applied Mathematics, vol. 25, SIAM, Philadelphia, 1976. | MR 0481301 | Zbl 0364.93002

Artstein Z. Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations 23 (1977), 224-243. (1977) | MR 0432985 | Zbl 0353.34044

Artstein Z. The limiting equations of nonautonomous ordinary differential equations, J. Differential Equations 25 (1977), 184-202. (1977) | MR 0442381 | Zbl 0358.34045

Artstein Z. Uniform asymptotic stability via the limiting equations, J. Differential Equations 27 (1978), 172-189. (1978) | MR 0466795 | Zbl 0383.34037

Coddington E.A.; Levinson N. Theory of ordinary differential equations, McGraw-Hill, New York, 1955. | MR 0069338 | Zbl 0064.33002

Davy J.L. Properties of the solution set of a generalized differential equation, Bull. Austral. Math. Soc. 6 (1972), 379-398. (1972) | MR 0303023 | Zbl 0239.49022

Engelking R. General Topology, PWN, Warsaw, 1977. | MR 0500780 | Zbl 0684.54001

Fedorchuk V.V.; Filippov V.V. General Topology. Basic Constructions (in Russian), Moscow University Press, Moscow, 1988.

Filippov V.V. On the theory of solution spaces of ordinary differential equations (in Russian), Dokl. Akad. Nauk SSSR 285 (1985), 1073-1077; English translation: Soviet Math. Dokl. 32 (1985), 850-854. (1985) | MR 0820601

Filippov V.V. On the ordinary differential equations with singularities in the right-hand side (in Russian), Mat. Zametki 38 (1985), 832-851; English translation: Math. Notes 38 (1985), 964-974. (1985) | MR 0823421

Filippov V.V. Cauchy problem theory for an ordinary differential equation from the point of view of general topology (in Russian), General Topology. Mappings of Topological Spaces, Moscow University Press, Moscow, 1986, 131-164. | MR 1080764

Filippov V.V. On stationary points and some geometric properties of solutions of ordinary differential equations (in Russian), Ross. Acad. Nauk Dokl. 323 (1992), 1043-1047; English translation: Russian Acad. Sci. Dokl. Math. 45 (1992), 497-501. (1992) | MR 1202308

Filippov V.V. On the Poincaré-Bendixon theorem and compact families of solution spaces of ordinary differential equations (in Russian), Mat. Zametki 53 (1993), 140-144. (1993) | MR 1220821

Hartman P. Ordinary differential equations, Wiley, New York, 1964. | MR 0171038 | Zbl 1009.34001

Kluczny C. Sur certaines familles de courbes en relation avec la théorie des équations différentielles ordinaires I, II, Annales Universitatis M. Curie-Skłodowska, Sec. A, Math. 15 (1961) 13-40; 16 (1962) 5-18.

Markus L. Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, vol. III, Annals of Math. Stud. 36, Princeton University Press, N.J., 1956, 17-29. | MR 0081388 | Zbl 0075.27002

Miller R.K. Asymptotic behavior of solutions of nonlinear differential equations, Trans. Amer. Math. Soc. 115 (1965), 400-416. (1965) | MR 0199502 | Zbl 0137.28202

Saks S. Theory of the integral, PWN, Warsaw, 1937. | Zbl 0017.30004

Savel'Ev P.N. On the Poincaré-Bendixon theorem and dissipativity in the plane (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1991, no. 3, 69-71; English translation: Moscow Univ. Math. Bull. 46 (1991), no. 3, 54-55. | MR 1204259

Sell G.R. Nonautonomous differential equations and topological dynamics I, II, Trans. Amer. Math. Soc. 127 (1967), 241-283. (1967) | MR 0212313

Strauss A.; Yorke J.A. On asymptotically autonomous differential equations, Math. Systems Theory 1 (1967), 175-182. (1967) | MR 0213666 | Zbl 0189.38502

Thieme H.R. Convergence results and a Poincaré-Bendixon trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (1992), 755-763. (1992) | MR 1175102

Yorke J.A. Spaces of solutions, Mathematical Systems Theory and Economics, Vol. II, Lecture Notes in Operations Research and Mathematical Economics, vol. 12, Springer-Verlag, New York, 1969, 383-403. | MR 0361294 | Zbl 0188.15502

Zaremba S.K. Sur certaines familles de courbes en relation avec la théorie des équations différentielles, Ann. Soc. Polon. Math. 15 (1936), 83-100. (1936)