A new criterion for controlling the number of limit cycles of some generalized Liénard equations
The authors consider the planar differential system \[ \dot x=p(x) +q(x)y, \qquad \dot y=r(x)+s(x)y+ t(x)y^*y, \tag{1} \] which includes the Liénard system \[ \dot x= y-f(x), \qquad \dot y=-g(x). \tag{2} \] The number of limit cycles of (1) is studied. The authors show that the problem can be reduced to the condition that a function, \(M(x)\), does not change sign. An application of the main result gives a sharp upper bound for the number of limit cycles of some systems (2), especially, yields exactly 3 limit cycles for the system \[ \dot x= y-x(x^*x-2)(x^*x-1) (x^*x-1/4), \qquad \dot y=-x. \tag{3} \] The reviewer points out that applying a previous result on the odd polynomial Liénard system to (3) gives immediately that (3) has at most 3 limit cycles [\textit{Y. Chen} and \textit{X. Chen}, J. Math. Res. Expo. 22, 368-370 (2002; Zbl 1074.34032)].
scientific article; zbMATH DE number 1884194
scientific article