## GENERALIZED TOPOLOGICAL TRANSITION MATRIX.

### Citations

320 |
Isolated invariant sets and Morse index
- Conley
- 1978
(Show Context)
Citation Context ...investigation. In particular, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See =-=[3]-=-, [4], [7], [8], [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in... |

56 | Connected simple systems and the Conley index of isolated invariant sets, Trans
- Salamon
- 1985
(Show Context)
Citation Context ... We assume that the reader is familiar with the basic ideas in Conley Index Theory, including Morse decompositions, homology index braids, connection matrices, etc. (see [3], [8], [9], [10], [16] and =-=[20]-=-). Let ϕ be a continuous flow on a locally compact Hausdorff space and let S be a compact invariant set under ϕ. A Morse decomposition of S is a collection of mutually disjoint compact invariant subse... |

38 |
The connection matrix theory for Morse decompositions
- Franzosa
- 1989
(Show Context)
Citation Context ...rticular, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], [4], [7], [8], =-=[9]-=-, [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in detecting global bi... |

25 |
Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory. Thesis (Ph.D
- Eidenschink
- 1995
(Show Context)
Citation Context ...nsition matrices in bifurcation-related results. See [3], [4], [7], [8], [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and =-=[6]-=-. Their continuation properties have proven useful in detecting global bifurcations. In particular, the continuation theorem [10] states that the connection matrices of an admissible ordering are inva... |

22 |
Index Filtrations and the Homology Index Braid for Partially Ordered Morse Decompositions
- Franzosa
- 1986
(Show Context)
Citation Context ...In particular, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], [4], [7], =-=[8]-=-, [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in detecting glob... |

17 |
Conley index, Handbook of dynamical systems
- Mischaikow, Mrozek
- 2002
(Show Context)
Citation Context ...sequence. We assume that the reader is familiar with the basic ideas in Conley Index Theory, including Morse decompositions, homology index braids, connection matrices, etc. (see [3], [8], [9], [10], =-=[16]-=- and [20]). Let ϕ be a continuous flow on a locally compact Hausdorff space and let S be a compact invariant set under ϕ. A Morse decomposition of S is a collection of mutually disjoint compact invari... |

16 |
The continuation theory for Morse decompositions and connection matrices
- Franzosa
- 1988
(Show Context)
Citation Context ...lar, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], [4], [7], [8], [9], =-=[10]-=- and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in detecting global bifurcat... |

16 |
Connected simple systems, transition matrices and heteroclinic bifurcations, Trans
- McCord, Mischaikow
- 1992
(Show Context)
Citation Context ...ey index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], [4], [7], [8], [9], [10] and =-=[14]-=-. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in detecting global bifurcations. In ... |

12 |
The connection map for attractor-repeller pairs
- McCord
- 1988
(Show Context)
Citation Context ...Smale flows without periodic orbits is unique for the flow-defined order. It is no surprise that the generalized topological transition matrix is unique. This is verified in Theorem 6. Furthermore in =-=[13]-=- and [21], an alternative and easier way to compute the connection matrix in this setting is presented. Likewise, we show in Theorem 6 that the generalized topological transition matrix can be compute... |

12 |
Connecting orbits in one-parameter families of flows. Ergodic Theory Dynam
- Reineck
(Show Context)
Citation Context ...for the introduction of transition matrices as a combinatorial mechanism to keep track of these changes. These transition matrices have since appeared in the literature under several guises: singular =-=[18]-=-, topological [14], and algebraic [11]. These three types of matrices are defined differently (particularly under contrasting conditions) and have distinct properties. On the other hand, due 2010 Math... |

11 |
The Conley Index and Floer
- Salamon
- 1990
(Show Context)
Citation Context ...ws without periodic orbits is unique for the flow-defined order. It is no surprise that the generalized topological transition matrix is unique. This is verified in Theorem 6. Furthermore in [13] and =-=[21]-=-, an alternative and easier way to compute the connection matrix in this setting is presented. Likewise, we show in Theorem 6 that the generalized topological transition matrix can be computed, withou... |

10 |
Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach
- Fiedler, Mischaikow
- 1992
(Show Context)
Citation Context ...ion. In particular, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], [4], =-=[7]-=-, [8], [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in detecting... |

9 |
Critical manifolds, travelling waves, and an example from population genetics
- Conley, Fife
- 1982
(Show Context)
Citation Context ...tigation. In particular, Conley index theory has proven to be quite useful in this role, as can be seen by the ample use of connection and transition matrices in bifurcation-related results. See [3], =-=[4]-=-, [7], [8], [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques [1], [2] and [6]. Their continuation properties have proven useful in dete... |

8 | Equivalence of topological and singular transition matrices in the Conley index - McCord, Mischaikow - 1995 |

6 |
The connection matrix in Morse-Smale flows
- Reineck
- 1990
(Show Context)
Citation Context ...e Morse decomposition consists of hyperbolic rest points and whenever the stable manifold of M(pi) and the unstable manifold of M(pi′) have nonempty intersection, it is transversal. As one can see in =-=[19]-=-, the connection matrix for Morse-Smale flows without periodic orbits is unique for the flow-defined order. It is no surprise that the generalized topological transition matrix is unique. This is veri... |

5 | Algebraic Transition Matrices in the Conley Index Theory
- Franzosa, Mischaikow
- 1998
(Show Context)
Citation Context ...g paper we will further explore properties of the generalized transition matrix and demonstrate how it generalizes all three transition matrices, namely, singular [18], topological [14] and algebraic =-=[11]-=-. In this paper we prove generalizations of the definition and properties of the classical topological transition matrices. With this in mind we restrict Definition 6 in order to obtain a new and broa... |

3 | conley: computing connection matrices in Maple - Barakat, Robertz |

2 |
Computation of connection matrices using the software package conley Internat
- Barakat, Maier-Paape
(Show Context)
Citation Context ...ction and transition matrices in bifurcation-related results. See [3], [4], [7], [8], [9], [10] and [14]. Connection matrices have been extensively studied and can be computed by numerical techniques =-=[1]-=-, [2] and [6]. Their continuation properties have proven useful in detecting global bifurcations. In particular, the continuation theorem [10] states that the connection matrices of an admissible orde... |

2 |
Spectral Sequences in Conley’s theory. Ergodic Theory and Dynamical Systems. Ergodic Theory Dynam
- Cornea, Rezende, et al.
(Show Context)
Citation Context ... Transition Matrix for the Sweeping Method. In this section we present an application of a generalized topological transition matrix in a continuation associated to a dynamical spectral sequence, see =-=[5]-=- and [12]. Our dynamical interpretation result implies the existence of connecting orbits in a fast-slow system “going from M(q) to M(p)” for a nontrivial entry on T rpq associated to the spectral seq... |

2 |
Continuation and Bifurcation Associated to the Dynamical
- Franzosa, Rezende, et al.
(Show Context)
Citation Context ...ion Matrix for the Sweeping Method. In this section we present an application of a generalized topological transition matrix in a continuation associated to a dynamical spectral sequence, see [5] and =-=[12]-=-. Our dynamical interpretation result implies the existence of connecting orbits in a fast-slow system “going from M(q) to M(p)” for a nontrivial entry on T rpq associated to the spectral sequence. Le... |

2 | de Rezende K.A., Silveira M.R. The convergence of Conley’s spectral sequence via the sweeping algorithm - Mello |