| Abstract: | In this paper, we consider the relative position of limit cycles for the system
$$\\begin{array}{*{20}{c}}
{\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}}
\end{array}\]$$
under the condition
$$\a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$
The main result is as follows:
(i)Under Condition (2), if $\\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\{(1)_{{\delta _0}}}\] $ has no limit cycles and
on singular closed trajectory through a saddle point in the whole plane,
(ii)Under condition (2), the foci 0 and R'' cannot be surrounded by the limit cycles of system (1) simultaneously. |
