Nonautonomous Conley index theory the connecting homomorphism

Axel Jänig

DOI: http://dx.doi.org/10.12775/TMNA.2019.006

Abstract


Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and strucuture of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.

Keywords


Nonautonomous differential equations; attractor-repeller decompositions; connecting orbits; connecting homomorphism; perturbations; semilinear parabolic equations; Morse--Conley index theory, partially orderd Morse-decompositions; nonautonomous Conley index, homology Conley index

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