Three-Dimensional Polynomial Differential Systems with an Isolated Compact Invariant Algebraic Surface

Abstract

The aim of this paper is to characterize the simplest three-dimensional polynomial differential system having an equilibrium and a 2-dimensional orientable smooth compact manifold with genus $g ≤ 1$ in $\mathbb{R}^3 ,$ where the 2-dimensional orientable smooth compact manifold is sphere $\mathbb{E}^ 2$ or torus $\mathbb{T}^2 $ We first look for the smallest degree of polynomial differential systems with both an equilibrium and an isolated compact invariant algebraic surface $\mathbb{E}^2$ or $\mathbb{T}^2.$ It is shown that the smallest degree of the system depends on the relative position between the equilibrium and the compact invariant algebraic surface in $\mathbb{R}^3.$ Furthermore, the sufficient and necessary algebraic conditions are given for the smallest order three-dimensional polynomial differential system having both an equilibrium and an isolated compact invariant algebraic surface. Lastly, we discuss the influence of the coexistence of an isolated compact invariant algebraic surface and an equilibrium on dynamics of the three-dimensional polynomial differential system.