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Topological properties of strongly monotone planar vector fields
Author(s) -
Balanov Z.,
Bolshakov A.,
Rachinskii D.
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800126
Subject(s) - mathematics , monotone polygon , fixed point , monotonic function , vector field , attractor , plane (geometry) , planar , pure mathematics , field (mathematics) , matrix (chemical analysis) , mathematical analysis , topology (electrical circuits) , combinatorics , geometry , computer science , computer graphics (images) , materials science , composite material
We consider strongly monotone continuous planar vector fields with a finite number of fixed points. The fixed points fall into three classes, attractors, repellers and saddles. Naturally, the relative positions of the fixed points must obey a set of restrictions imposed by monotonicity. The study of these restrictions is the main goal of the paper. With any given vector field, we associate a matrix describing the arrangement of the fixed points on the plane. We then use these matrices to formulate simple necessary and sufficient conditions which allow one to determine whether a finite set of attractors, repellers and saddles at given positions on the plane can be realized as the fixed point set of a strongly monotone vector field.