Explicit Expression for First Integral of a Rational Type of Kolmogorov Systems

International Journal of Mathematics Trends and Technology

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Volume 12 | Number 2 | Year 2014 | Article Id. IJMTT-V12P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V12P511

Explicit Expression for First Integral of a Rational Type of Kolmogorov Systems

Khalil I.T. Al-Dosary

Abstract

In this paper we cherecterize the integrability and introduce an explicit expression of rst integral then consequently the non-existence of pe- riodic orbits for rational type of the planar Kolmogorove systems of the form x = x(Pn1 (x,y) /Pn2 (x,y) + Rk1 (x,y)/Rk2 (x,y)) y = y (Qm1 (x,y)/Qm2 (x,y) + Rk1(x,y)/Rk2 (x,y)) where n1, n2, m1, m2, k1, and k2 are positive integers and Pi , Qj and Rk are homogeneous polynomials of degree i , j and k respectively such that n1 - n2 = m1 - m2. We also present an example in order to illustrate the applicability of the result.

Keywords

Intergability, Kolmogorov system, Periodic orbit, Rational function

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Citation :

Khalil I.T. Al-Dosary, "Explicit Expression for First Integral of a Rational Type of Kolmogorov Systems," International Journal of Mathematics Trends and Technology (IJMTT), vol. 12, no. 2, pp. 69-74, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V12P511