The global dynamics of a class of vector fields in \(\mathbb R^3\)
This paper is devoted to the investigation of the two vector fields \[ X(x)=(x_1+f_1(x))\frac{\partial}{\partial x_1}(x) + (x_2+f_2(x))\frac{\partial}{\partial x_2}(x) + (x_3+f_3(x))\frac{\partial}{\partial x_3}(x) \] and \[ Y(x)=x+f(x)= (x_1+f_1(x), x_2+f_2(x), x_3+f_3(x)), \] where \(x=(x_1, x_2, x_3)\in\mathbb R^3,\) and \(f_i (x)\,(i=1,2,3)\) are homogeneous polynomials of degree \(n\) in \(\mathbb R^3\). The authors establish conditions such that all the possible nonwandering sets of the vector fields fall into three classes, namely singularities; closed orbits; graphs of unions of singularities and the trajectories connecting them. Moreover, criteria of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of the vector fields and some applications of these results are given.
scientific article; zbMATH DE number 6145917