Abstract
In this paper, we define the Conley index $${\mathfrak{h}(D)}$$ for a region of discontinuity D of a piecewise C k discontinuous vector field Z on an n-dimensional compact Riemannian smooth orientable manifold and prove it to be a homotopy invariant. This invariance is obtained by regularization of the discontinuous vector field. We use an adapted form of Lyapunov graph continuation to produce, in a few examples, a regularization of the discontinuous vector field with the property that the dynamics in a regularized neighborhood of D has the same Conley index as $${\mathfrak{h}(D)}$$ .
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