Nonautonomous Conley index theory the connecting homomorphism

Axel Jänig



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.


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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P. Brunovsky and P. Polacik, The Morse–Smale structure of a generic reactiondiffusion equation in higher space dimensions, J. Differential Equations 135 (1997), 129–181.

M.C. Carbinatto and K.P. Rybakowski, Homology index braids in infinite-dimensional Conley index theory, Topol. Methods Nonlinear Anal. 26 (2005), 35–74.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer–Verlag Berlin, Heidelberg, New York, IV, 1981, 348 p.

A. Jänig, Nonautonomous conley index theory: Continuation of Morse decompositions, Topol. Methods Nonlinear Anal. 53 (2019), 57–77.

A. Jänig, The Conley index along heteroclinic solutions of reaction-diffusion equations, J. Differential Equations 252 (2012), 4410–4454.

A. Jänig, A non-autonomous Conley index, J. Fixed Point Theory Appl. 19 (2017), 1825–1870.

A. Jänig, Nonautonomous Conley index theory: The homology index and attractor-repeller decompositions, Topol. Methods Nonlinear Anal., DOI: 10.12775/TMNA.2018.040.

M. Mrozek and K.P. Rybakowski, A cohomological Conley index for maps on metric spaces, J. Differential Equations 90 (1991), 143–171.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.

K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, 1987.

G.R. Sell and Y. You, Dynamics of evolutionary equations., Appl. Math. Sci., vol. 143, Springer, New York, 2002, xiii+670 p.

G.R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold, 1971.

E.H. Spanier, Algebraic topology, McGraw–Hill, New York, 1966.


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