Christopher McCord

On the global dynamics of attractors for scalar delay equations

A semi-conjugacy from the dynamics of the global attractors for a family of scalar delay differential equations with negative feedback onto the dynamics of a simple system of ordinary differential equations is constructed. The construction and proof are done in an abstract setting, and hence, are valid for a variety of dynamical systems which need not arise from delay equations. The proofs are based on the Conley index theory. INSTITUTE FOR DYNAMICS, UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO 45221-0025 E-mail address: chris .mccorduc .edu CENTER FOR DYNAMICAL SYSTEMS AND NONLINEAR STUDIES, GEORGIA INSTITUTE OF TECH- NOLOGY, ATLANTA, GEORGIA 30332 E-mail address: mischaikfmath.gatech.edu This content downloaded from 157.55.39.27 on Wed, 07 Sep 2016 05:19:39 UTC All use subject to http://about.jstor.org/terms

Connected simple systems, transition matrices and heteroclinic bifurcations

Given invariant sets A, B, and C, and connecting orbits A → B and B → C, we state very general conditions under which they bifurcate to produce an A → C connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets

Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds

McCord (1991) claimed that Nielsen coincidence numbers and Lefschetz coincidence numbers are related by the inequality N(f,g) 2 IL(f,g)l f or all maps f, g : S1 --t SZ between compact orientable solvmanifolds of the same dimension. It was further claimed that N(f, g) = lL(f, g)l when ,572 is a nilmanifold. A mistake in that paper has been discovered. In this paper, that mistake is partially repaired. A new proof of the equality N(f, g) = ]~5(f, g)l for nilmanifolds is given, and a variety of conditions for maps on orientable solvmanifolds are established which imply the inequality N(f, 9) 2 lL(f, g)l. H owever, it still remains open whether N(f, g) 2 IL(f, g)l for all maps between orientable solvmanifolds.

Poincaré-Lefschetz duality for the homology Conley index

The Conley index for continuous dynamical systems is defined for (one-sided) semiflows. For (two-sided) flows, there are two indices defined: one for the forward flow; and one for the reverse flow. In general, the two indices give different information about the flow; but for flows on orientable manifolds, there is a duality isomorphism between the homology Conley indices of the forward and reverse flows. This duality preserves the algebraic structure of many of the constructions of the Conley index theory: sums and products; continuation; attractor-repeller sequences and connection matrices.

Mappings and homological properties in the Conley index theory

The role in the Conley index of mappings between flows is considered. A class of maps is introduced which induce maps on the index level. With the addition of such maps to the theory, the homology Conley index becomes a homology theory. Using this structure, an analogue of the Lefschetz theorem is proved for the Conley index. This gives a new condition for detecting fixed points of flows, extending the classical Euler characteristic condition.

The integral manifolds of the three body problem

Introduction The decomposition of the spaces The cohomology The analysis of

An integrated machine vision based system for solving the nonconvex cutting stock problem using genetic algorithms

{\mathfrak K}(c,h)

Planar central configuration estimates in the n -body problem

for equal masses The analysis of

Are Hamiltonian flows geodesic flows

{\mathfrak K}(c,h)

The three faces of Nielsen: Coincidences, intersections and preimages

for general masses Bibliography.