Roman Srzednicki

CHAPTER 7 – Ważewski Method and Conley Index

This chapter discusses the retract method introduced by Tadeusz Wazwski. It is a method of proving the existence of solutions that remain in a given set and refers to differential equations describing some evolution in time. The sets under consideration should satisfy the condition that all egress points are strict or its less restrictive variant. Now they are called Wazewski sets. The method is based on theorems that roughly assert that there is a solution contained a Wazewski set for all positive values of time if the subset of egress points is not a retract of the whole set. If, moreover, the set is compact, then its invariant part is nonempty. For isolating blocks , that is, compact Wazewski sets that do not contain any full solutions intersecting their boundaries, Charles Conley discovered homotopical invariant, which provides quantitative information on their invariant parts. It is called the Conley index . The chapter describes the Wazewski method in detail and presents information on foundations of the Conley index theory, which directly relates to the method and essentially does not overlap with the exposition of Mischaikow and Mrozek.

A generalization of the Lefschetz fixed point theorem and detection of chaos

We consider the problem of existence of fixed points of a continuous map f X -* X in (possibly) noninvariant subsets. A pair (C, E) of subsets of X induces a map f t: C/E -* C/E given by ft([x]) = [f(x)] if x, f(x) e C \ E and ft ([x]) = [E] elsewhere. The following generalization of the Lefschetz fixed point theorem is proved: If X is metrizable, C and E are compact ANRs, and ft is continuous, then f has a fixed point in C \ E provided the Lefschetz number of H* (ft) is nonzero. Actually, we prove an extension of that theorem to the case of a composition of maps. We apply it to a result on the existence of an invariant set of a homeomorphism such that the dynamics restricted to that set is chaotic.

The Conley index over a base

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in R3, and the problem of continuing two periodic orbits which are not homotopic as loops.

A generalized Lefschetz fixed point theorem and symbolic dynamics in delay equations

We prove a generalized version of the Lefschetz fixed point theorem, and use it to obtain a variety of periodic and aperiodic solutions for differential delay equations; in particular, of the type \dot x(t) = f(x(t-1)) . Here f:\mathbb R \to \mathbb R is odd and two-periodic, and we obtain both strictly periodic solutions and solutions periodic modulo a multiple of two. The qualitative behavior of solutions can be coded by symbol sequences containing the sequence of levels about which these solutions oscillate.

On time-duality of the Conley index

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Generalized Lefschetz theorem and a fixed point index formula

\rm\check H^{k} (h(s))\cong H_{dim(M)-k}({h}^*(s))

A Topological Approach to the Algorithmic Computation of the Conley Index for Poincaré Maps

where h(S) denotes the Conley index of an isolated invariant set S in a given flow on an oriented manifold M and h*(S) denotes the index of S with respect to the same flow with the time parameter reversed.

On isolated sets of solutions of some two-point boundary value problems☆

A new algorithm for obtaining rigorous results concerning the existence of chaotic invariant sets of dynamical systems generated by non-autonomous, time-periodic differential equations is presented. Unlike all other algorithms the presented algorithm does not require the numerical integration of the solutions and as a consequence it is insensitive to the rapid error growth in the case of long integration. The result is based on a new theoretical approach to the computation of the homology of the Poincaré map. A concrete numerical example concerning a time-periodic differential equation in the complex plane is provided.