ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

The Asymptotic Behavior of Non-extendable Solutions of Quasi-homogeneous Systems


Yuri Vasilievich Churin

Dept. of differential equation
Faculty of Mathematics and Mechanics
Saint-Petersburg State University


This paper explores non-autonomous systems of ordinary differential equations. Outside of some sphere in phase space the right side of the system is close to an autonomous homogeneous system which has no bounded solutions and generates a Morse-Smale dynamical system on the Poincare sphere. We study the asymptotic behavior of inextensible solutions and prove that in this case the solution norm increases with no limit, and its projection onto the unit sphere tends to a non-wandering orbit of the Morse-Smale system.



  1. Pliss V. A. [On the number of periodic solutions to equations with polynomial right -hand side. ] Doklady Akademii nauk SSSR, 1959; 127 ( 5): 965-968. (In Russ. )
  2. Churin Yu. V. [ On disappearance of periodic solutions of of quasi-homogeneous systems with only simple exclusive multiple. ] Differetsialnye uravneniya, 1975, XI, No 4, 678-686. (In Russ. )
  3. Churin Yu. V. [Behavior solutions of quasi-homogeneous system in the neighborhood simple exclusive destinations . ] Nelinejnye dinamicheskie systemy [Nonlinear dynamic systems]. St. Petersburg: Publishing House of St. Petersburg University, 1997, Issue 1, pp. 298-311.
  4. Churin Yu. V. [Simple exceptional many uncontained of quasi-homogeneous Systems. ] Differentsialnye uravneniya, 1973, IX, No. 6, 1073-1084. (In Russ. )
  5. Pilyugin S. Yu. Vvedenie v grubye systemy differentsialnykh uravnenij [Introduction to gross systems of differential equations. ] Leningrad, Publishing House of the Leningrad University, 1988. 188 p. (In Russ. )

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