Nonautonomous Conley index: theory and applications

Conley index theory associates isolated invariant sets with an index e.g, a topological space. This theory is generalized to nonautonomous dynamical systems. A corresponding theory of Morse-decompositions is developed. Applications to dynamical systems obtained from ordinary and partial differential equations are discussed

Nonautonomous Conley index theory. The homology index and attractor-repeller decompositions

In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution operators. Different variants such as a categorial index, a homotopy index and a homology index are obtained. Furthermore, attractor-repeller decompositions and conecting homomorphisms are introduced for the nonautonomous setting

Conley index orientations

The homotopy Conley index along heteroclinic solutions of certain parabolic evolution equations is zero under appropriate assumptions. This result implies that the so-called connecting homomorphism associated with a heteroclinic solution is an isomorphism. Hence, using $\mathbb{Z}$-coefficients it can be viewed as either $1$ or $-1$ - depending on the choice of generators for the homology Conley index. We develop a method to choose such generators, and compute the connecting homomorphism relative to these generators

Der Morse-Komplex für Reaktions-Diffusionsgleichunge

The singular homology of a compact smooth Riemannian manifold can be described by means of its Morse-Smale-Witten chain complex. There are proofs of this which rely on Conley index theory. We generalize these ideas to cover a class of semilinear parabolic equations, notably reaction-diffusion equations. Finally, one obtains a Morse complex for suitable isolated invariant sets.Der Morse-Smale-Witten Kettenkomplex beschreibt die singuläre Homologie kompakter glatter Riemannscher Mannigfaltigkeiten. Dies lässt sich durch Verwendung der Conley-Indextheorie beweisen. Die zugrundeliegenden Zusammenhänge werden verallgemeinert, und es ergibt sich ein Morse-Komplex für invariante Mengen bestimmter semilinearer parabolischer Differentialgleichungen, insbesondere Reaktions-Diffusionsgleichungen