Existence and uniqueness and stability of limit cycle for a class of planar nonlinear systems

In this paper, a class of planar nonlinear differential systems ẋ = y(1+sin²mx), ẏ = -x+δy+axy+bx³+cx²y+λx⁴e^x2y is studied. By the formal series method based on Poincaré ideas, the center and the focus are judged, and by the Dulac function, the non-existence of closed orbits is discussed. Meantime, by the Hopf bifurcation theory, some sufficient conditions for the existence of limit cycles which bifurcate from the equilibrium point are analyzed, then by some proper transforms, and by the theorem of L.A.Cherkas and L.I.Zheilevych, some sufficient conditions for the uniqueness and stability of limit cycles for such systems are established. Finally, one example is given for illustration.