arXiv subjects

Joji Benny

Publications and source records attributed to Joji Benny.

2 recordsLinked to original sources

arXiv2023

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

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 $\mathbb{S}^2$.

math.DS↗

arXiv2022

Invariant Hyperplane Sections of Vector Fields on the Product of Spheres

Let $S_{p,q}$ be the hypersurface in $\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \left\lbrace (x_1,\ldots,x_{p+1},x_{p+2},\ldots,x_{p+q+1}) \in \mathbb{R}^{p+q+1} \big| \left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 + \sum_{j=p+2}^{p+q+1} x_j^2 = 1 \right\rbrace,$$ where $a > 1$. We show that $S_{p,q}$ is homeomorphic to the product $S^p \times S^q$. We classify all degree one and two polynomial vector fields on $S_{p,q}$. We consider the polynomial vector field $\mathcal{X} = (R_1,...,R_{p+1},R_{p+2},...,R_{p+q+1})$ in $\mathbb{R}^{p+q+1}$ which keeps $S_{p,q}$ invariant. Then we study the number of certain invariant algebraic subsets of $S_{p,q}$ for the vector field $\mathcal{X}$ if either $p>1$ or $q>1$.

math.DS↗