Topological properties of strongly monotone planar vector fields

Z. Balanov, A. Bolshakov and D. Rachinskii

Mathematische Nachrichten, 2019, vol. 292, issue 10, 2108-2128

Abstract: 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.

Date: 2019
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https://doi.org/10.1002/mana.201800126

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