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Unformatted text preview: Page 1 Invariant Manifolds and Liapunov Functions Invariant Manifolds The motivation for invariant manifolds comes from the study of critical elements of linear differential equations of the form ˙ x = Ax, x ∈ R n . Let E s , E c , and E u be the (generalized) real eigenspaces of A associated with eigenvalues of A lying on the open left half plane, the imaginary axes, and the open right half plane, respectively. As we have seen in the section on linear systems, each of these spaces is invariant under the flow of ˙ x = Ax and represents, respectively, a stable, center, and unstable subspace. Let us call a subset S ⊂ R n a k-manifold if it can be locally represented as the graph of a smooth function defined on a k-dimensional affine sub- space of R n . As in the calculus of graphs, k manifolds have well defined tangent spaces at each point and these are independent of how the mani- folds are represented as graphs. Although the notion of a manifold is much more general, this will serve our purposes. A k-manifold S ⊂ R n is said to be invariant under the flow of a vector field X if for x ∈ S , F t ( x ) ∈ S for small t > 0; i.e., X is tangent to S . One can thus say that an invariant manifold is a union of (segments of) integral curves . Invariant manifolds are intuitively “nonlinear eigenspaces.” A little more precisely, we may define invariant manifolds S of a critical element γ ; that is, γ is a fixed point or a periodic orbit, to be stable or unstable depending 2 on whether they are comprised of orbits in S that wind toward γ with increasing, or with decreasing time. Let us focus on fixed points, say, x e to begin. In a neighborhood of x e , the tangent spaces to the stable and unstable manifolds are provided by the generalized eigenspaces E s , E c , and E u of the linearization A = DX ( x e ). We are going to start with hyperbolic points ; that is, points where the linearization has no center subspace. Let the dimension of the stable subspace be denoted k . Theorem (Local Invariant Manifold Theorem for Hyperbolic Points) . As- sume that X is a smooth vector field on R n and that x e is a hyperbolic equilibrium point. There is a k- manifold W s ( x e ) and a n- k-manifold W u ( x e ) each containing the point x e such that the following hold: i. Each of W s ( x e ) and W u ( x e ) is locally invariant under X and con- tains x e . ii. The tangent space to W s ( x e ) at x e is E s and the tangent space to W u ( x e ) at x e is E u . iii. If x ∈ W s ( x e ) , then the integral curve with initial condition x tends to x e as t → ∞ and if x ∈ W u ( x e ) , then the integral curve with initial condition x tends to x e as t → -∞ ....
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- Fall '09
- Stability theory, Center manifold, Center Manifold Theorem, J. E. Marsden, M. A. Golubitsky