摘要
In this paper, we discuss the boundedness of the solutions, the existence andthe uniqueness of the limit cycle of the following cubic differential system:x’=y, y’=-x+δy+a<sub>2</sub>xy+a<sub>4</sub>x+a<sub>5</sub>x<sup>2</sup>y. （*）We obtain the following results:（1） System （*） is bounded if and only if （i） a<sub>5</sub>【0, a<sub>4</sub>=0; or （ii） a<sub>5</sub>=0, a<sub>4</sub>【0, δ≤0,-（-8a<sub>4</sub>）<sup>1/2</sup>【a<sub>2</sub>【（-8a<sub>4</sub>）<sup>1/2</sup>.（2） System （*） has no limit cycle if a<sub>5</sub>δ≥0.（3） System （*） has one and only one limit cycle if a<sub>5</sub>δ【0, for a<sub>4</sub>≤0.
In this paper, we discuss the boundedness of the solutions, the existence andthe uniqueness of the limit cycle of the following cubic differential system:x'=y, y'=-x+δy+a_2xy+a_4x+a_5x^2y. (*)We obtain the following results:(1) System (*) is bounded if and only if (i) a_5<0, a_4=0; or (ii) a_5=0, a_4<0, δ≤0,-(-8a_4)^(1/2)<a_2<(-8a_4)^(1/2).(2) System (*) has no limit cycle if a_5δ≥0.(3) System (*) has one and only one limit cycle if a_5δ<0, for a_4≤0.
基金
This paper is supported by the China Youth Natural Science Foundation.
