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

On the Acyclicity of a Cubic Differential System Having Four Invariant Straight Lines and a Unstable Focus

Author(s):

Adam Damirovich Ushho

Adygea State University, Department of Theoretical Physics
Associate Professor, Candidate of Physical and Mathematical Sciences,
208 Pervomayskaya str., Maykop, Russian Federation

uschho76@rambler.ru

Abstract:

The sufficient conditions for the absence of limit cycles in a cubic autonomous differential system having four invariant straight lines and a unstable focus are found. Examples illustrating the provisions of the acyclicity theorem of a cubic system with four invariant straight lines and a unstable focus are given. Examples of cubic differential systems with four invariant lines having a limit cycle located both inside and outside the parallelogram formed by invariant straight lines are considered.

Keywords

• cubic autonomous differential system
• invariant straight line
• limit cycle
• non-rough focus
• off-knot point

References:

1. Bautin N. N. [On Periodic Solutions of a System of Differential Equations]. Prikladnaya Matematika i Mekhanika, 1954, no. 1, p. 128. (In Russ. )
2. Belevec P. S., Valeeva R. T. [On a class of dynamical systems with two partial algebraic integrals]. Volzhskij matematicheskij sbornik, 1973, no. 16, pp. 52-54. (In Russ. )
3. Belevec P. S., Valeeva R. T. [Qualitative picture of a whole class of dynamic systems]. Volzhskij matematicheskij sbornik, 1973, no. 16, pp. 55-60. (In Russ. )
4. Stolyarov V. V. [On limit cycles and limitations of one dynamic system]. Differencial'nye uravneniya, 1989, no. 10, pp. 1823-1824. (In Russ. )
5. Ushkho D. S., Teshev R. M. [Limit cycles of one cubic differential system]. Vestnik Adygejskogo gos. un-ta, 2002, no. 3-4 (7-8), pp. 139-143. (In Russ. )
6. Lyubimova R. A. [On a differential equation with integral straight lines]. Differencial'nye i integral'nye uravneniya, 1977, no. 1, pp. 19-22. (In Russ. )
7. Gorbuzov V. N., Tyshchenko V. YU. [Partial integrals of systems of ordinary differential equations]. Matematicheskij sbornik, 1992, no. 3, pp. 76-94. (In Russ. )
8. Ushkho D. S., Ushkho A. D. [On the existence of limit cycles and linear partial integrals of cubic differential systems on the plane]. Trudy FORA, 2004, no. 9, pp. 20-24. (In Russ. )
9. Bautin N. N., Leontovich E. A. Metody i priemy kachestvennogo issledovaniya dinamicheskih sistem na ploskosti [Methods and techniques of qualitative research of dynamical systems on the plane]. Moscow, Nauka Publ., 1976. 496 p. (In Russ. )
10. Ushkho D. S. i dr. Ob odnom metode issledovaniya chisla invariantnyh pryamyh polinomial'nyh vektornyh polej n-oj stepeni [On a method for investigating the number of invariant straight lines of the n-th order polynomial vector fields]. Vestnik Adygejskogo gos. un-ta, 2018, no. 4 (231), pp. 15-27. (In Russ. )
11. Andronov A. A., Leontovich E. A., Gordon I. I., Majer A. G. Teoriya bifurkacij dinamicheskih sistem na ploskosti [Theory of bifurcations of dynamical systems on a plane]. Moscow, Nauka Publ., 1967. 488 p. (In Russ. )
12. Otrokov N. F. Analiticheskie integraly i predel'nye cikly [Analytical integrals and limit cycles]. Gor'kij: Volgo-Vyatskoe knizhnoe izdatel'stvo, 1972. 216 p. (In Russ. )
13. Cherkas L. A. [On the absence of limit cycles of one equation having a rough focus]. Differencial'nye uravneniya, 1970, no. 5, p. 779-783. (In Russ. )

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