Invariant Circles and Phase Portraits of Cubic Vector Fields on the Sphere

被引：2

作者：

Benny, Joji ^{[1
]}

Jana, Supriyo ^{[1
]}

Sarkar, Soumen ^{[1
]}

机构：

[1] Indian Inst Technol Madras, Dept Math, Chennai, India

来源：

QUALITATIVE THEORY OF DYNAMICAL SYSTEMS | 2024年 / 23卷 / 03期

关键词：

Polynomial vector fields; Kolmogorov system; Periodic orbit; Invariant circle; Invariant great circle; First integral; Phase portrait; INTEGRABILITY; DYNAMICS; SYSTEMS;

D O I：

10.1007/s12346-024-00979-0

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere S-2 = {(x, y, z) E R-3|x(2) + y(2) +z(2) = 1}. We start by classifying all degree three polynomial vector fields on S2 and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on S-2 and also study the maximum number of various types of invariant great circles for homogeneous cubic vector fields on S-2. We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on S-2.

引用

页数：23

共 19 条

[1] EQUILIBRIA AND DYNAMICS IN AN ECONOMIC PREDATOR-PREY MODEL OF AGRICULTURE [J].

APEDAILLE, LP ;

FREEDMAN, HI ;

SCHILIZZI, SGM ;

SOLOMONOVICH, M .

MATHEMATICAL AND COMPUTER MODELLING, 1994, 19 (11) :1-15

[2] Multiplicity of invariant algebraic curves in polynomial vector fields [J].

Christopher, Colin ;

Llibre, Jaume ;

Pereira, Jorge Vitorio .

PACIFIC JOURNAL OF MATHEMATICS, 2007, 229 (01) :63-117

[3]

Dumortier F, 2006, UNIVERSITEXT, P1

[4]

Gasull A, 1996, J MATH ANAL APPL, V198, P25, DOI 10.1006/jmaa.1996.0065

[5]

Jana S, 2023, Arxiv, DOI arXiv:2307.09439

[6]

Kolmogorov A., 1936, GIORNALE DELLISTITUT, V7, P74

[7] Limit cycle oscillations, response time, and the time-dependent solution to the Lotka-Volterra predator-prey model [J].

Leconte, M. ;

Masson, P. ;

Qi, Lei .

PHYSICS OF PLASMAS, 2022, 29 (02)

[8]

Llibre J., 2006, Extracta Math., V21, P167

[9]

Llibre J., 2008, B ACAD TIIN E REPUB, V1, P19

[10] Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere [J].

Llibre J. ;

Pessoa C. .

Rendiconti del Circolo Matematico di Palermo, 2006, 55 (1) :63-81

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