Bifurcation Problems In Nonlinear System And Center Manifold Theorem For Flows

Bifurcation problems play an important part in nonlinear dynamical systems, we now give a definition to it for the needs of research in the future. Consider the following systems: x=fυ(x); x∈Rn,υ∈Rl. (1) This is a system of differential equations depending on the l-dimensional parameter u, and the solutions of the equation fυ(x)=0 are the equilibrium solutions of(1). If the flow of (1) is not structurally stable at a point u=υO, then we sayυO is a bifurcation value of u.From the definition given above, we find out that the bifurcation problems are closely related with the structure of flows. In fact, if the stable manifolds and unstable manifolds intersect non-transversally, the structure of dynamical system must be unstable.Now, we introduce Hartman-Grobman's theorem to state the relation of the linear sys-tems and the nonlinear systems.Theorem 1 Suppose O is a hyperbolic fixed point of the nonlinear system, then there exist a neighborhood U of O, in which f(x) is topologically equivalent to Df(O)x.In fact, the conclusion could be more accurate:Theorem 2 (Stable Manifold Theorem) Suppose that x=f(x) has a hyperbolic fixed point O, then there exist local stable and unstable manifoldsWlocs(0), Wlocu(O), of the same dimension ns, nυas those of the Es,Eu of the linearized system, and tangent to Es, E" at O.If at the origin, the linearization of f has no pure imaginary eigenvalues, then as the Stable Manifold Theorem shows, the topological equivalence of the flow is determined by the number of eigenvalues with positive and negative real parts. If there are eigenvalues with zero parts, then, we can find out that near the origin, the flow could always be very complicated.Through the analysis of the above, we eventually get the most important theorem:Theorem 3 (Center Manifold Theorem for Flows)If f is a Cr vector field vanishing at the origin. Then at the origin, there exist Cr stable manifolds Ws tangent to the stable subspace, Crunstable manifolds Wu tangent to the unstable subspace and Cr center manifolds Wc tangent to the center subspace. Here, stable manifolds Ws, unstable manifolds Wand center manifolds Wc are all invariant. The stable manifolds Ws and unstable manifolds Wu are unique, but the center manifolds Wc is not necessarily unique.For simplicity, and also because it is an interesting case in physics, from now on, we always assume the unstable manifold is empty and the linear part of the bifurcating system is in block diagonal form: (x,y, u)∈Rn×Rm×Rl, the "parameter" u plays a role of a dependent variable, where B(u) and C(u) are n×n and m×m matrices, whose eigenvalues have, respectively, zero real parts and negative real parts, and f and g vanish, along with their first partial derivatives, at the point(x, y, u)= (0,0,0).Because Wc is tangent to the center subspace (y=0), then we can define it as follows: Wc={(x,y,u)}y=h(x,u), h(0,0)=Dh(0,0)=0} (3) h:U→Rm,where U is a neighborhood of the origin in Rn+l.We now consider the projection of the vector field on y=h(x, u) onto the center sub-space:Theorem 4 If at the origin, the solutions of equation (4)is locally asymptotically stable(resp. unstable), then at the origin, the solutions of equation (2)is also locally asymp-totically stable(resp. unstable).