Abstract
In the present paper we investigate the absence, for dynamical systems of :he first degree of structural instability on two-dimensional compact orientable manifolds of any genus, of nonclosed Poisson-stable semitrajectories and trajectories that are doubly asymptotic to a double limit cycle. The propositions are some of the basic propositions that must be added to the known conditions for the first degree of structural instability on a plane (or sphere) [1] in order to obtain a description of dynamical systems of the first degree of structural instability on orientable two-dimensional manifolds. Systems of the first degree of structural instability on a torus were considered in [2]. A study of such systems on two-dimensional manifolds of higher genus has not been carried out up to the present time.
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