Approximation Of Center Manifolds In General Nonlinear Systems

Dynamical systems theory have a wide range applications in chemistry, physics, eco-nomics, ecology, control theory and numerical calculations. In this paper we consider the l-dimensional nonlinear systems F(x) is Cr(r≥1) vector field and the origin O is an isolated equilibrium point, ie F(0)=0. Among them we introduce the concepts of center manifold on the system, the stable manifold theorem and the center manifold theorem. This paper focus on the case which the unstable manifold does not exist on the system (1).Consider the system (x, y)∈Rm×Rn, A and B are m-dimensional and n-dimensional constant matrices, The real part of the eigenvalues in A are0, which in B are negatives,f and g both are C2functions, and f(0,0)=0, Df(0,0)=0, g(0,0)=0, Dg(0,0)=0.Generally speaking, the center manifold of system(2) can not be solved exactly, but we can get any of its precision approximate solution. Scaling this system, we can get Let and T means transposition, flp= yp=y1P1y2P2…ynPn,|l|=l1+l2+…+lm,|p|=p1+p2+…pnLet and which implies<λ,l)=Σk=1m λklk,e1,e2,…en are nature base.The main result in this paper:When|x|is sufficiently small,Γε is the uniformly valid one-order approximation of the center manifold of system(3) at equilibrium point(0,0).