The number and linking of periodic solutions of periodic systems (Q793208)

The author considers the differential system (1) \(dx/dt=f(t,x)\), \(t\in R\), \(x\in R^ 2\) where f is assumed to be a \(C^ 1-R^ 2-valued\) function which is periodic in t of period 1 and such that for each \((t_ 0,x_ 0)\in R\times R^ 2\) the solution \(x=\phi(t;t_ 0,x_ 0)\) exists for all \(t\in(-\infty,+\infty)\). It is further assumed that there is a closed set in \(R^ 2\) which is homeomorphic to a closed disk and is invariant under the Poincaré transformation. The main result states that the equation has periodic solutions of every integer period provided that there are three periodic solutions of period 1 which are attractor or repeller and link together in a certain kind of complicated manner.