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InvariantManifolds

InvariantManifolds - Page 67 Invariant Manifolds There are...

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Page 67 Invariant Manifolds There are two basic motivations for invariant manifolds. The Frst comes from the notion of separatrices that we have seen in our study of planar systems, as in the Fgures. We can ask what is the higher dimensional gen- eralization of such separatrices. Invariant manifolds provides the answer. The second comes from our study of of linear systems: ˙ 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 our study of linear systems, each of these spaces is invariant under the ±ow of ˙ x = Ax and represents, respectively, a stable, center, and unstable subspace. We want to generalize these notions to the case of nonlinear systems. Thus,

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68 Introduction invariant manifolds will correspond, intuitively, to “nonlinear eigenspaces.” Let us call a subset S R n a k -manifold if it can be locally represented as the graph of a smooth function deFned on a k -dimensional a±ne sub- space of R n . As in the calculus of graphs, k manifolds have well deFned tangent spaces at each point and these are independent of how the mani- folds are represented (or parametrized) 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 ²ow of a vector Feld X if for x S , F t ( x ) S for small t> 0, where F t ( x ) is the ²ow of X . One can show that this is equivalent to the condition that X is tangent to S . One can thus say that an invariant manifold is a union of (segments of) integral curves of X . While one can study invariant manifolds associated to general invariant sets, such as periodic orbits, let us focus on Fxed points, say, x e to begin— these correspond to the origin for a linear system. There will be three sorts of invariant manifolds, namely stable manifolds , center mani- folds , and unstable manifolds . In a neighborhood of x e , the tangent spaces to the stable, center, 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 Feld 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 .
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InvariantManifolds - Page 67 Invariant Manifolds There are...

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