Singular Point Assignment via the Conley Index Theory

Singular boundary value problems via the Conley index

We use Conley index theory to solve the singular boundary value problem Duxx+ f(u, ux, x) = 0 on an interval [−1, 1], where u ∈ R n and D is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley index theory is used here not for detection of an invariant set, but for tracking ce...

Connecting Fast-slow Systems and Conley Index Theory via Transversality

Geometric Singular Perturbation Theory (GSPT) and Conley Index Theory are two powerful techniques to analyze dynamical systems. Conley already realized that using his index is easier for singular perturbation problems. In this paper, we will revisit Conley’s results and prove that the GSPT technique of Fenichel Normal Form can be used to simplify the application of Conley index techniques even ...

The Conley Index, Gauge Theory, and Triangulations

This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.

Conley Index Theory and Novikov-Morse Theory

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on thes...

Equivalence of Topological and Singular Transition Matrices in the Conley Index Theory