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...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...

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... 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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... 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...

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...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...