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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Periodic solutions of periodic retarded functional differential equations

Pawłowski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

353-363

The paper presents a geometric method of finding periodic solutions of retarded functional differential equations (REDE) x'(t) = f(t,x1), where f is T-periodic in t. We construct a pair of subsets of R x R^n called a T-periodic block and compute its Lefschetz number. If it is nonzero, then there exists a T-periodic solution.

Optimal periodic orbits for non-recurrent rational functions

Urbański M.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

175-184

We prove that each non-parabolic periodic orbit contained in the omega-limit set of a measure-recurrent, optimal orbit for a continuous function defined on the Julia set of a non-recurrent rational function is also optimal. As a by-product, we prove in the next section appropriate versions of shadowing and closing lemmas for non-recurrent rational functions.