The Conley index for discontinuous vector fields

Author(s)

Casagrande, R. ^[1] ; de Rezende, K. A. ^[1] ; Teixeira, M. A. ^[1]

Affiliation

^[1] Univ Estadual Campinas, UNICAMP, Dept Math, BR-13083970 Campinas, SP - Brazil

• Document type: Journal article
• Source: Geometriae Dedicata; v. 136, n. 1, p. 47-56, OCT 2008.
• Web of Science Citations: 0

Abstract

In this paper, we define the Conley index 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 h(D). (AU)

• FAPESP's process: 02/10246-2 - Control systems, dynamical systems, stochastic dynamical systems, Lie theory and differential geometry
• Grantee: Luiz Antonio Barrera San Martin
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 04/10229-6 - Algebraic, geometric and differential topology
• Grantee: Daciberg Lima Gonçalves
• Support Opportunities: Research Projects - Thematic Grants