Casagrande, R.; De Rezende, K. A.; Teixeira, M. A. The conley index for discontinuous vector fields. (English) Zbl 1151.37305 Geom. Dedicata 136, 47-56 (2008). Summary: 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 \({\mathfrak{h}(D)}\). Cited in 1 Document MSC: 37B30 Index theory for dynamical systems, Morse-Conley indices 34A36 Discontinuous ordinary differential equations Keywords:Conley index theory; discontinuous equations; vector fields Software:conley PDFBibTeX XMLCite \textit{R. Casagrande} et al., Geom. Dedicata 136, 47--56 (2008; Zbl 1151.37305) Full Text: DOI References: [1] Andronov A.A., Vitt A.A., Khaikin S.E.: Theory of Ocillators. Dover, New York (1996) · Zbl 0188.56304 [2] di Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Olivar, G., Piiroinen, P.T.: Bifurcations in Non-smooth Dynamical Systems. Publications of the Bristol Centre for Applied Nonlinear Mathematics, N.4 (2005) · Zbl 1168.34006 [3] Bertolim, M.A., Mello, M.P., De Rezende, K.A.: Poincaré-Hopf and Morse Inequalities for Lyapunov Graphs, vol. 25, pp. 1–39. Ergodic Theory and Dynamical Systems (2005) · Zbl 1084.37013 [4] Broucke M.E., Pugh C.C, Simić S.N.: Structural stability of piecewise smooth systems. Comput. Appl. Math. 20(1–2), 51–89 (2001) · Zbl 1121.37307 [5] Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS Lecture Notes, vol. 38. A.M.S. Providence, R.I. (1978) · Zbl 0397.34056 [6] Fillipov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer (1988) [7] Gottlieb D.H., Samaranayake G.: The index of discontinuous vector fields. NY J. Math. 1, 130–148 (1995) · Zbl 0883.57025 [8] Kunze, M.: Non-smooth Dynamical Systems. Lecture notes in Mathematics, 1744. Springer, Berlin-New York (2000) · Zbl 0965.34026 [9] Mrozek M.: A cohomological index of Conley type for multi-valued admissible flows. J. Differ. Equ. 84, 15–51 (1990) · Zbl 0703.34019 [10] Sotomayor, J., Teixeira, M.A.: Regularization of discontinuous vector fields. In: Magalhães L., Rocha C., Sanches L. (eds.) Proceedings of the International Conference on Differential Equations, Equadiff 95, Lisboa, pp. 207–223. World Scientific (1998) · Zbl 0957.37015 [11] Sotomayor J., Machado A.F.: Structurally stable discontinuous vector fields in the plane. Qual. Theory Dyn. Syst. 3, 227–250 (2002) · Zbl 1047.37011 [12] Teixeira M.A.: Stability conditions for discontinuous vector fields. Int. Conf. J. Diff. Equ. 88, 15–29 (1990) · Zbl 0780.34035 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.