Nonautonomous Conley index theory the connecting homomorphism
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 indexAbstract
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.References
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