The connection matrix in Morse-Smale flows

Reineck, James F.

doi:10.1090/S0002-9947-1990-0972705-3

The connection matrix in Morse-Smale flows
HTML articles powered by AMS MathViewer

by James F. Reineck

Trans. Amer. Math. Soc. 322 (1990), 523-545

DOI: https://doi.org/10.1090/S0002-9947-1990-0972705-3

PDF | Request permission

Abstract:

In a Morse-Smale flow with no periodic orbits, it is shown that the connection matrix is unique. In the case of periodic orbits, nonuniqueness can occur. We show that on $2$-manifolds, with some technical assumptions, given a connection matrix for the flow, one can replace the periodic orbits with doubly-connected rest points and obtain a new flow with no periodic orbits having the given connection matrix.

References

Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, DOI 10.1002/cpa.3160370204
John M. Franks, Morse-Smale flows and homotopy theory, Topology 18 (1979), no. 3, 199–215. MR 546790, DOI 10.1016/0040-9383(79)90003-X

Index filtrations and connection matrices for partially ordered Morse decompositions

Robert Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193–213. MR 857439, DOI 10.1090/S0002-9947-1986-0857439-7
Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. MR 978368, DOI 10.1090/S0002-9947-1989-0978368-7
Robert D. Franzosa, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), no. 2, 781–803. MR 973177, DOI 10.1090/S0002-9947-1988-0973177-6
Christopher McCord, The connection map for attractor-repeller pairs, Trans. Amer. Math. Soc. 307 (1988), no. 1, 195–203. MR 936812, DOI 10.1090/S0002-9947-1988-0936812-4

Classification of travelling wave solutions to reaction-diffusion equations

Predator-prey with group defense

a connection matrix approach

S. Newhouse and M. M. Peixoto, There is a simple arc joining any two Morse-Smale flows, Trois études en dynamique qualitative, Astérisque, No. 31, Soc. Math. France, Paris, 1976, pp. 15–41. MR 0516405
James F. Reineck, Travelling wave solutions to a gradient system, Trans. Amer. Math. Soc. 307 (1988), no. 2, 535–544. MR 940216, DOI 10.1090/S0002-9947-1988-0940216-8
Dietmar Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044, DOI 10.1090/S0002-9947-1985-0797044-3