Genericity for Non-Wandering Surface Flows

Consider the set χ nw 0 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 c...

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Bibliographic Details
Published in:Journal of dynamical and control systems 2017-04, Vol.23 (2), p.197-212
Main Author: Yokoyama, Tomoo
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Continuity (mathematics)
Control
Dynamical Systems
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Orbital stability
Systems Theory
Toruses
Vibration
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Summary:Consider the set χ nw 0 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 nw 0 ; (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 nw 0 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 .
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-015-9303-6