Flows without wandering points on compact connected surfaces

Cobo, Milton; Gutiérrez, Carlos; Llibre, Jaume

doi:10.1090/s0002-9947-10-05113-5

Cited by 19 publications

(22 citation statements)

References 12 publications

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“…, U N make a finite covering of the surface. Considering l ′ smaller we can also suppose that orbits of the orbit segments in ∂U i do not meet l ′ , call it condition (2). Let a i be the local cross section of ∂U i where the flow enters to the flow box.…”

Section: Wandering Pointsmentioning

confidence: 99%

Expansive flows of surfaces

Artigue¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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Section: Wandering Pointsmentioning

confidence: 99%

Expansive flows of surfaces

Artigue¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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“…Thus Sing(v) is finite. By Theorem 3 [3], each singular point is either a center or a multi-saddle. Finally, we will show that v has no multi-saddles.…”

Section: Lemmamentioning

confidence: 97%

“…Thus we may assume that Sing(v) is finite. By Theorem 3 [3], each singular point is either a center or a multi-saddle. By Corollary 2.9 [10], the closed subset Sing(v) P consists of finitely many orbits.…”

Section: Introductionmentioning

confidence: 97%

Genericity for Non-Wandering Surface Flows

Yokoyama

2015

J Dyn Control Syst

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Consider the set χ 0 nw of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M −Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in χ 0 nw ; (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in χ 0 nw if and only if the surface is homeomorphic to either the sphere S 2 , the projective plane P 2 , or the Klein bottle K 2 .

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“…Then, Cobo et al [11] have shown that they have topological centers, k-saddles and k/2-∂-saddles. A stagnation point p is called a topological center, if there is a neighborhood of p where all orbits except p are periodic.…”

Section: Structurally Unstable Hamiltonian Vector Fieldsmentioning

confidence: 99%

“…This is not an easy task, since the transient streamline pattern is structurally unstable and it may thus contain many singular streamline structures [11]. In the present study, we propose a combinatorial method providing a list of possible transitions between two structurally stable streamline topologies from their word representations.…”

Section: Introductionmentioning

confidence: 99%

Transitions between streamline topologies of structurally stable Hamiltonian flows in multiply connected domains

Sakajo

Yokoyama

2015

Physica D: Nonlinear Phenomena

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h i g h l i g h t s• A combinatorial procedure providing a list of possible transitions between streamline topologies. • It is applicable to any physical phenomena described by 2D Hamiltonian vector fields.• It provides many global and generic transitions between streamline topologies. • It is applicable to snapshots of streamline patterns observed experimentally. • A new data compression algorithm for a large amount of long-time flow evolution data. a b s t r a c tWe consider Hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two-dimensional multiply connected domains. One example of such Hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow, whose Hamiltonian is called the stream function. Here, we are concerned with topological structures of the level sets of the Hamiltonian, which we call streamlines by analogy from incompressible fluid flows. Classification of structurally stable streamline patterns has been considered in Yokoyama and Sakajo (2013), where a procedure to assign a unique sequence of words, called the maximal word, to these patterns is proposed. Thanks to this procedure, we can identify every streamline pattern with its representing sequence of words up to topological equivalence. In the present paper, based on the theory of word representations, we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any Hamiltonian. Although this method cannot deal with topological streamline changes induced by bifurcations, it reveals the existence of many non-trivial global transitions in a generic sense. We also demonstrate how the present theory is applied to fluid flow problems with vortex structures.

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Flows without wandering points on compact connected surfaces

Abstract: Abstract. Given a compact 2-dimensional manifold M we classify all continuous flows ϕ without wandering points on M . This classification is performed by finding finitely many pairwise disjoint open ϕ−invariant subsets

Cited by 19 publications

References 12 publications

Expansive flows of surfaces

Expansive flows of surfaces

Genericity for Non-Wandering Surface Flows

Transitions between streamline topologies of structurally stable Hamiltonian flows in multiply connected domains

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