ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On the Problems of Distinguishing Exceptional Directions of a Two-dimensional System at a Singular Point


A. F. Andreev

St.Petersburg State University,
Chair of Differential Equations,
Bibliotechnaya sq., 2, 198904, St.Petersburg, Russia


On the plane of real variables x,y , we consider an autonomous quasihomogeneous (algebroidal) system of differential equations for which O=(0,0) is an isolated singular point. We discuss the question of detecting all curves "tangent" to O ( called TO-curves or characteristic curves), i.e. semitrajectories entering O along certain directions. The problem is to give unified general conditions on perturbations (i.e. on terms of higher order as compared with main homogeneous terms of the system) under which perturbations do not change a topological type of the behaviour of trajectories of the initial homogeneous system a) in a neighbourhood of any characteristic (exceptional) direction at O, or b) in a neighbourhood of every exceptional direction of a certain type. These conditions are stated in terms of an order of smoothness of perturbations in a neighborhood D of O and of an order of their smallness as (x,y) tends to (0,0). By examples, these conditions are shown to be close to necessary ones.

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