Geometriae Dedicata

, 136:47 | Cite as

The Conley index for discontinuous vector fields

  • R. Casagrande
  • K. A. de RezendeEmail author
  • M. A. Teixeira
Original Paper


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 \({\mathfrak{h}(D)}\).


Conley index theory Discontinuous vector fields 

Mathematics Subject Classification (2000)

37B30 34A36 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • R. Casagrande
    • 1
  • K. A. de Rezende
    • 1
    Email author
  • M. A. Teixeira
    • 1
  1. 1.Department of MathematicsIMECC/C.P.6065, University of Campinas–UNICAMPCampinasBrazil

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