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+a2xy+a4x+a5x2y. (*)We obtain the following results:(1) System (*) is bounded if and only if (i) a50, a4=0; or (ii) a5=0, a40, δ≤0,-(-8a4)1/2a2(-8a4)1/2.(2) System (*) has no limit cycle if a5δ≥0.(3) System (*) has one and only one limit cycle if a5δ0, for a4≤0.
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