Invariant closed surface and stability of non-hyperbolic equilibrium point for polynomial differential systems in R3 ✩

In this paper, by algebraic method and Lyapunov function, we discuss the stability of non-hyperbolic equilibrium point in R3, that the coefficient matrix of linearized system have a pair purely imaginary eigenvalues and a zero eigenvalue, with the perturbations of 3th-degree homogeneous and 3th-degree and 5th-degree homogeneous. We shall give the sufficiently conditions which can immediately distinguish that the equilibrium point is asymptotically stable or unstable and a ball-center by the coefficients of perturbed terms, meantime, we discuss the condition which produce invariant closed surface by changing the stability of equilibrium point with perturbation. © 2006 Elsevier Inc. All rights reserved.