BSU Bulletin. Mathematics, Informatics
Bibliographic description:
ON THE NUMBER OF PERIODIC SOLUTIONS FOR SOME POLYNOMIAL DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS // BSU Bulletin. Mathematics, Informatics. - 2020. №1. . - С. 28-34.
Title:
ON THE NUMBER OF PERIODIC SOLUTIONS FOR SOME POLYNOMIAL DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS
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We have considered the differential equations (generalized Abel equations), which right-hand sides are polynomials of degree n>2 with continuous coefficients that periodically depend on the argument. Many authors estimated the number of periodic solutions for such equations. In this article, it is assumed that the coefficient for the highest degree of polynomial is either non-negative or non- positive, and some of the coefficients are zero. In the case of odd n it is proved that the equation has at most three periodic solutions and their multiplicity is analyzed. In the case of even n and zero free term of the right-hand side, it has been established that the equation has at most four periodic solutions and the multiplicity of these solutions is also analyzed.
Keywords:
polynomial differential equation with periodic coefficients; Abel equation; number of periodic solutions.
List of references:
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Neto A. L. On the Number of Solutions of the Equation for which x(0)=x(1). In- vent. Math, 1980, vol. 59, no. 2, pp. 67–76.
Ylyashenko Ju. Hilbert-Type Number for Abel Equations, Growth and Zeros of Holomorfic Functions. Nonlinearity. 2000. Vol. 13. Pp. 1337–1342.
Casull A., Guillamon A. Limit Cycles for Generalized Abel Equations. J. Bifur- cation Chaos. 2006. Vol. 16. Pp. 3737–3745.
Ivanov G. G., Alferov G. V., Korolev V. S., Selitskaya E. A. Periodicheskie resheniya differentsialknykh uravnenij [Periodic Solutions for Diffe- rential Equations]. Vestnik Permskogo universiteta. Matematika. Mekhanika. Infor- matika. 2019. No. 2(46). Pp. 5–15.
Neto A. L. On the Number of Solutions of the Equation for which x(0)=x(1). In- vent. Math, 1980, vol. 59, no. 2, pp. 67–76.
Ylyashenko Ju. Hilbert-Type Number for Abel Equations, Growth and Zeros of Holomorfic Functions. Nonlinearity. 2000. Vol. 13. Pp. 1337–1342.
Casull A., Guillamon A. Limit Cycles for Generalized Abel Equations. J. Bifur- cation Chaos. 2006. Vol. 16. Pp. 3737–3745.
Ivanov G. G., Alferov G. V., Korolev V. S., Selitskaya E. A. Periodicheskie resheniya differentsialknykh uravnenij [Periodic Solutions for Diffe- rential Equations]. Vestnik Permskogo universiteta. Matematika. Mekhanika. Infor- matika. 2019. No. 2(46). Pp. 5–15.
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