Three-Dimensional Polynomial Differential Systems with an Isolated Compact Invariant Algebraic Surface (Journal of Nonlinear Modeling and Analysis, 2025)
The aim of this paper is to characterize the simplest three-dimensional polynomial differential system having an equilibrium and a 2-dimensionalorientable 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 $ Wefirst look for the smallest degree of polynomial differential systems with bothan 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 relativeposition between the equilibrium and the compact invariant algebraic surfacein $\mathbb{R}^3.$ Furthermore, the sufficient and necessary algebraic conditions are givenfor the smallest order three-dimensional polynomial differential system havingboth an equilibrium and an isolated compact invariant algebraic surface. Lastly, we discuss the influence of the coexistence of an isolated compact invariantalgebraic surface and an equilibrium on dynamics of the three-dimensionalpolynomial differential system.