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

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

## Equilibrium States and Adjacent Questions of the Plane Polynomial Vector Fields Theory

### Author(s):

Vyacheslav Beslanovitch Tlyachev

Russia, 385000, Maykop, Pervomayskaya str., 208
Head of the Department of Theoretical Physics, Engineering Physics Faculty

stvb2006@rambler.ru

Russia, 385000, Maykop, Pervomayskaya str., 208
Associate Professor of the Department of Theoretical Physics,
Engineering Physics Faculty

uschho76@rambler.ru

Damir Salikhovitch Ushho

Russia, 385000, Maykop, Pervomayskaya str., 208
Associate Professor of the Department of Mathematical Analysis
Faculty of Mathematics and Computer Sciences

damirubych@mail.ru

### Abstract:

The general theorems on the equilibrium states of autonomous dynamical systems whose right-hand side are n-th degree polynomials are proved. In particular, it is shown that if a system whose right-hand side are mutually simple polynomials has a set number of equilibrium states, then they are all simple. Moreover, when certain conditions for the relations between the right parts of the system are met, the Poincare index of any equilibrium state is one less than the power of the polynomial. The conditions of the absence of limit cycles of a cubic system having special points of the «center» type are considered. Using the canonical form of a cubic system having the maximum number of equilibrium states, Poincare indices are determined, which allows us to judge their types. Examples are provided to support the claims made.

### Keywords

• center-focus problem
• cubic system
• equilibrium states
• isoclines
• Poincare index
• Poincare sphere
• polynomial vector field

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