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Invariant circles and phase portraits of cubic vector fields on the sphere

Joji Benny, Supriyo Jana, Soumen Sarkar

Abstract

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

Invariant circles and phase portraits of cubic vector fields on the sphere

Abstract

In this paper, we characterize and study dynamical properties of cubic vector fields on the sphere . We start by classifying all degree three polynomial vector fields on and determine which of them form Kolmogorov systems. Then, we show that there exist completely integrable cubic vector fields on and also study the maximum number of various types of invariant circles for homogeneous cubic vector fields on . We find a tight bound in each case. Further, we also discuss phase portraits of certain cubic Kolmogorov vector fields on .
Paper Structure (5 sections, 13 theorems, 83 equations, 3 figures)

This paper contains 5 sections, 13 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cubic vector fields on $\mathbb{S}^2$
  4. Cubic homogeneous vector fields on $\mathbb{S}^2$
  5. Cubic Kolmogorov vector fields on $\mathbb{S}^2$

Key Result

Proposition 2.1

LlMe07 Let $\mathcal{X}$ be a polynomial vector field in $\mathbb{R}^n$ and $W$ a finite dimensional vector subspace of $\mathbb{R}[x_1, x_2, \ldots , x_n]$ with $\dim(W) > 1$. If $f=0$ is an invariant algebraic hypersurface for the vector field $\mathcal{X}$ with $f\in W$, then $f$ is a factor of $

Figures (3)

  • Figure 1: Phase portraits when neither (a) nor (b) of \ref{['eq:a_and_b']} is true.
  • Figure 2:
  • Figure 3: When $A = 5$ and $B = -1$, and $C = 2$.

Theorems & Definitions (26)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • ...and 16 more