Periodic Solutions of Periodic Systems

Abstract

In this chapter we study the existence, stability and isolation of periodic solutions belonging to n-dimensional systems of periodic nonlinear differential equations of the form ẋ = f (t, x) where f is periodic in t with some period T > 0: f (t + T, x) = f (t,x). One may believe that an autonomous system of the form ẋ = f (x) is a special case since, obviously, here the right-hand side is periodic in t with arbitrary positive period. Though this is true, autonomous systems cannot be treated similarly to periodic non-autonomous ones. This is so because in the case of an autonomous system we do not know a priori what may be the period of a periodic solution if there exists any, and also because the integral curve belonging to a non-constant periodic solution of an autonomous system can never be “isolated”. More will be said about these problems at the appropriate places. We have mentioned these problems here in order to explain why autonomous systems will be treated in the next chapter. The methods developed with the aim of establishing the existence and stability of periodic solutions can be classified in two groups. The first is the group of topological methods based on degree theory and fixed point theorems. These methods will be presented in the first Section of this chapter. The background material can be found in Appendix 2. The second group consists of (small) perturbation methods. These are more effective but have the disadvantage that they work under the assumption that the given differential equation is a “perturbation” of another one whose periodic solution is known. Both methods have their origin in the works of H. Poincaré [1899]. We shall treat the perturbation methods separately in Chapter 6. In the second section of this chapter we study the stability and isolation problems of periodic solutions. In Sections 3, 4 and 5, applications will be presented.