Periodic perturbation of planar systems with a semistable limit cycle (Q1370172)

Concerning the unperturbed two-dimensional differential system \[ dx/dt =f(x) \tag{*} \] it is assumed that (*) has a semistable limit cycle \(L\) of multiplicity two. The author considers the perturbed system \[ dx/dt =f(x)+ \varepsilon g(t,x, \varepsilon) \tag{**} \] where \(f\) and \(g\) are \(C^3\) with respect to their variables, and \(g\) is periodic in \(t\) with period \(2\pi\). By means of the averaging method, the method of invariant manifolds and the Melnikov technique, the author derives sufficient conditions for (**) to have two invariant tori or no invariant torus, and subharmonic solutions.