Generalized topological transition matrix

Robert Franzosa, Ketty A. de Rezende, Ewerton R. Vieira



This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this more general transition matrix satisfies the additional requirement that it covers flow-defined Conley-index isomorphisms, one proves algebraic and connection-existence properties. These general transition matrices with this covering property are referred to as generalized topological transition matrices and are used to consider connecting orbits of Morse-Smale flows without periodic orbits, as well as those in a continuation associated to a dynamical spectral sequence.


Conley index; connection matrices; transition matrices; Morse-Smale system; sweeping method; spectral sequence

Full Text:



M. Barakat and S. Maier-Paape, Computation of connection matrices using the software package conley Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), no. 9, 3033–3056.

M. Barakat and D. Robertz, Conley: computing connection matrices in Maple, J. Symbolic Comput. 44 (2009), no. 5, 540–557.

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. Amer. Math. Soc., Providence, R.I., 1978. iii+89 pp.

C. Conley and P. Fife, Critical manifolds, travelling waves, and an example from population genetics, J. Math. Biol. 14 (1982), no. 2, 159–176.

O. Cornea, K.A. de Rezende and M.R. Silveira, Spectral Sequences in Conley’s theory. Ergodic Theory and Dynamical Systems, Ergodic Theory Dynam. Systems 30 (2010), no. 4, 1009–1054.

M. Eidenschink, Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory, Ph.D. Thesis, Georgia Institute of Technology, 1995, 215 pp.

B. Fiedler and K. Mischaikow, Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach ,Arch. Rational Mech. Anal. 119 (1992), no. 2, 145–196.

R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193–213.

R. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592.

R. Franzosa, The continuation theory for Morse decompositions and connection matrices. Trans. Amer. Math. Soc. 310 (1988), no. 2, 781–803.

R. Franzosa and K. Mischaikow, Algebraic transition matrices in the Conley index theory, Trans. Amer. Math. Soc. 350 (1998), no. 3, 889–912.

R. Franzosa, K.A. de Rezende and M.R. Silveira, Continuation and Bifurcation Associated to the Dynamical Spectral Sequence, Ergodic Theory Dynam. Systems 334 (2014), 1849–1887.

T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka and J. Reineck, The Conley index for fast-slow systems I: One dimensional slow variables, J. Dynam. Differential Equations 11 (1999), 427–470.

H. Kokubu, On transition matrices, EQUADIFF99, Proceedings of the International Conference on Differential Equations, Berlin, Germany 1–7 August 1999, World Scientific, 2000, pp.219–224, 133–144.

H. Kokubu, K. Mischaikow and H. Oka, Directional Transition Matrix, Banach Center Publications, 47, 1999.

C. McCord, The connection map for attractor-repeller pairs, Trans. Amer. Math. Soc. 307 (1988), no. 1, 195–203.

C. McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), no. 1, 397–422.

C. McCord and K. Mischaikow, Equivalence of topological and singular transition matrices in the Conley index theory, Michigan Math. J. 42 (1995), no. 2, 387–414.

K. Mischaikow and M. Mrozek, Conley Index, Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 393–460.

M. Mello, K.A. de Rezende and M.R. Silveira, The convergence of Conley’s spectral sequence via the sweeping algorithm, Topology Appl. 157, (2010), 2111–2130.

T. Moeller, Conley–Morse Chain Maps, Thesis, Georgia Institute of Technology, August 2005.

J. F. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory Dynam. Systems 8∗ (1988), Charles Conley Memorial Issue, 359–374.

J. F. Reineck, The connection matrix in Morse–Smale flows, Trans. Amer. Math. Soc. 322 (1990), no. 2, 523–545.

D. A. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41.

D. A. Salamon, Morse theory, The Conley index and Floer homology, Bull. London Math. Soc. 22 (1990), no. 2, 113–140.


  • There are currently no refbacks.

Partnerzy platformy czasopism