The Conley index for discontinuous vector fields / O indice de Conley para campos de vetores descontinuos
AUTOR(ES)
Rogerio Casagrande
DATA DE PUBLICAÇÃO
2008
RESUMO
The Conley index is a used as a topological invariant in the analysis of the qualitative behavior of dynamical systems. lnitially the theory was developed for continuous flows in finite dimensional spaces and later extended to the infinite dimensional setting as well as to the discrete case. ln this work, we present a Conley index theory for a class of discontinuous vectar fields, with discontinuity of the first kind. Discontinuous vector fields are frequent in several areas of Science and Engineering and can be expressed as piecewise differentiable vector fields on an n-dimensional compact manifold, M. We consider a Whitney sttatification of M and a discontinuous vector field Z on M, where the region of discontinuity, D, is the strata of codimension one. We show the existence of a D-index pair (N, L) and prove that the quotienL space N/L independs on the pair chosen, thus defining the D-Conley index as the homotopy type of this quotient space. We present some examples of its calculation. We also use the D-Conley index to show sufficient conditions for the existence of bifurcation points in a one parameter family of discontinuous vector fields. We also present a theory of continuation for Lyapunov D-graphs associated to this class of discontinuous vector fields.
ASSUNTO(S)
sistemas dinamicos topologia algebrica discontunuous vector fields conley index theory algebraic topology campos vetoriais teoria do indice de conley dynamical systems
