Bifurcations of topological type at singular points of parametrized vector fields

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

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ON POLYNOMIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON A CIRCLE WITHOUT SINGULAR POINTS

Bulletin of the South Ural State University series Mathematics Mechanics Physics ◽

10.14529/mmph200404 ◽

2020 ◽

Vol 12 (4) ◽

pp. 33-40

Author(s):

V.Sh. Roitenberg ◽

Keyword(s):

Differential Equations ◽

Limit Cycles ◽

Vector Fields ◽

Second Order ◽

Singular Points ◽

Free Term ◽

Small Perturbations ◽

Cylindrical Phase ◽

Autonomous Differential Equations ◽

The Right

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

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RESEARCH OF STATIONARY PLANE FIELDS BY MEANS OF COMPLEX POTENTIAL

METHODS AND DEVICES OF QUALITY CONTROL ◽

10.31471/1993-9981-2019-1(42)-39-46 ◽

2019 ◽

pp. 39-46

Author(s):

V. P. Nisonskii ◽

Yu. V Kornuta ◽

I. M-B. Katamai

Keyword(s):

Field Theory ◽

Complex Variable ◽

Complex Potential ◽

Vector Fields ◽

Oil And Gas ◽

Water Reservoir ◽

Singular Points ◽

Vector Analysis ◽

Variable Theory ◽

Complex Variable Theory

Some of the most frequently encountered and generic types of plane vector fields with singular points at the origin of the coordinate system have been studied using complex variable theory methods combined with complex potential methods and field theory methods. The basic concepts of field theory and vector analysis, which are used to study vector fields and the main numerical characteristics of these fields, have been considered. The study of the most frequently encountered vector fields with singular points of four types, namely the generator, the vortical point, the eddy source, the dipole, have been conducted. The application of the complex potential for finding the main characteristics of the vector fields of the considered types, namely their divergence and rotor, has been shown. Equipotential lines and streamlines of the considered vector fields have been obtained and graphically constructed using the method of complex potential. Studied using the vector analysis methods and the methods of the theory of complex variable functions (complex potential) characteristics of vector fields can be used for mathematical modeling of various problems, arising during the study of layers, namely soil and water reservoir filtration problems, as well as in studying the flow of fluid or gas in layers problems. The developed and considered mathematical models of flat vector fields and the found numerical characteristics of these fields can be used to solve other problems of the oil and gas complex, which require studies of the flow of liquids or gases in gas- or oil-bearing beds.

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