Invariant closed surface and stability of non-hyperbolic equilibrium point for polynomial differential systems in \(\mathbb R^3\) (Q865349)

The author considers a nonlinear differential equation of the form \[ \dot{x}=Ax + p(x), \] where \(A\in\mathbb{R}^{3\times 3}\) has the eigenvalues \(\{i,-i,0\}\) and is in real Jordan normal form. The nonlinear perturbation \(p\) is a polynomial either of degree three or of degree five with some special structure. Sufficient conditions for the coefficients of the polynomial \(p\) are given, for which the trivial solution \(x\equiv 0\) is asymptotically stable, stable, or unstable, respectively. Furthermore, conditions for the existence of invariant solution surfaces are studied. The results are proved with the help of the very simple Lyapunov-function \(V(x)=x\cdot x^T\), \(x\in\mathbb{R}^3\). The results seem to be only a minor contribution to the stability theory of nonlinear differential equations. It is not clear why the author restricted himself to the three-dimensional state space, since most, if not all, results seem to carry over easily to the general \(n\)-dimensional state space. Although the paper is well structured the reading is difficult because the English is very bad. Finally, there seems to be too less references to existing literature.