Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point
Abstract: The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6k + 1, k ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.
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In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form x ˙ = x ( R n - 1 ( x , y ) + P n ( x , y ) + S n + 1 ( x , y ) ) , y ˙ = y ( R n - 1 ( x , y ) + Q n ( x , y ) + S n + 1 ( x , y ) ) , \matrix{ {\dot x = x\left( {{R_{n - 1}}\left( {x,y} \right) + {P_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr {\dot y = y\left( {{R_{n - 1}}\left( {x,y} \right) + {Q_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr } where n is a positive integer, Rn−1 , Pn , Qn and Sn +1 are homogeneous polynomials of degree n − 1, n, n and n + 1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.
In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form x ˙ = x ( R n - 1 ( x , y ) + P n ( x , y ) + S n + 1 ( x , y ) ) , y ˙ = y ( R n - 1 ( x , y ) + Q n ( x , y ) + S n + 1 ( x , y ) ) , \matrix{ {\dot x = x\left( {{R_{n - 1}}\left( {x,y} \right) + {P_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr {\dot y = y\left( {{R_{n - 1}}\left( {x,y} \right) + {Q_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr } where n is a positive integer, Rn−1 , Pn , Qn and Sn +1 are homogeneous polynomials of degree n − 1, n, n and n + 1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.
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