Necessary conditions for a limit cycle and its basin of attraction

In this paper we consider a general differential equation of the form x=f (x) with f epsilon C-l (R-n, R-n) and n greater than or equal to 2. Borg, Hartman, Leonov and others have studied sufficient conditions for the existence, uniqueness and exponential stability of a periodic orbit and for a set to belong to its basin of attraction. They used a certain contraction property of the flow with respect to the Euclidian or a Riemannian metric. In this paper we also prove sufficient conditions including upper bounds for the Floquet exponents of the periodic orbit. Moreover, we show the necessity of these conditions using Floquet theory and a Lyapunov function.