Directional transition matrix

Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, Hiroe

doi:10.4064/-47-1-133-144

Cited by 3 publications

(1 citation statement)

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“…The simplest construction of a transition matrix is due to Reineck [11] and is referred to as the singular transition matrix. Similar to the case of a connection matrix, an off-diagonal non-zero entry of the transition matrix implies the occurrence of bifurcations (see also Kokubu et al [4] and McCord and Mischaikow [6,7]).…”

Section: Introductionmentioning

confidence: 97%

Tangencies and the Conley index

Arai¹

2002

Ergod. Th. Dynam. Sys.

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We construct an algebraic method for detecting bifurcations of discrete dynamical systems using the Conley index theory. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. As an application, we give a sufficient condition for the occurrence of homoclinic and heteroclinic tangencies by showing that the occurrence of a heterodimensional cycle in the projectivization of a dynamical system implies the tangency of the original system.

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Section: Introductionmentioning

confidence: 97%

Tangencies and the Conley index

Arai¹

2002

Ergod. Th. Dynam. Sys.

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Propagation of hexagonal patterns near onset

et al. 2003

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Dedicated to Klaus Kirchgässner on the occasion of his seventieth birthday with our deep gratitude for his encouragement, mentorship and support.For a pattern-forming system with two unbounded spatial directions that is near the onset to instability, we prove the existence of modulated fronts that connect (i) stable hexagons with the unstable trivial pattern, (ii) stable hexagons with unstable roll solutions, (iii) stable hexagons with unstable hexagons, and (iv) stable roll solutions with unstable hexagons. Our approach is based on spatial dynamics, bifurcation theory, and geometric singular perturbation theory.

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Transition matrix theory

Franzosa

Vieira

2017

Trans. Amer. Math. Soc.

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Abstract. In this article we present a unification of the theory of algebraic, singular, topological and directional transition matrices by introducing the (generalized) transition matrix which encompasses each of the previous four. Some transition matrix existence results are presented as well as verification that each of the previous transition matrices are cases of the (generalized) transition matrix. Furthermore we address how applications of the previous transition matrices to the Conley Index theory carry over to the (generalized) transition matrix.Dedicated to the memory of James Francis Reineck

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Directional transition matrix

Abstract: Abstract. We present a generalization of topological transition matrices introduced in [6].

Cited by 3 publications

References 10 publications

Tangencies and the Conley index

Tangencies and the Conley index

Propagation of hexagonal patterns near onset

Transition matrix theory

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