23 lines
1.6 KiB
Markdown
23 lines
1.6 KiB
Markdown
**Case 1**: if the critical point is hyperbolic, life is okay. Linearize about that point, look at eigenvalues and eigenvectors to understand our different manifolds.
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**Case 2:** If the point is NOT hyperbolic. We've got to do something else.
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Assume $\vec{P} \in R^3$ is a critical point in our system $\dot{X} = F(x), x\in R^3$
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Define stable and unstable manifolds of that point P as:
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$$ W_s(\vec{P}) = \left\{x: \Lambda^+(x) = \vec{P} \right\}$$
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$$ W_u(\vec{P}) = \left\{x: \Lambda^-(x) = \vec{P} \right\}$$
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Where the first is forward in time, the second is backward in time.
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**Theorem:** x is some some differential equation system in R^n and $f = c^1(E)$ (c1 continuous over E, where E is an open subset of R^n, containing the origin)
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If $f(0)=0$, the Jacobian has n eigenvalues with a nonzero real part. (Hyperbolic)! Then in a small neighborhood of $x\approx 0$ There exists stable and unstable manifolds of the linearized system $$\dot{x} = Jx$$
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where $J$ is the Jacobian, and $W_s$ and $W_u$ are tangent to $E_s$ and $E_u$ respectively at $x=0$. E defines the eigenspace.
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What they hell do we do when eigenvalues do not have a real part?
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**Center Manifold:** $W_c$ and **Center Eigenspace:** $E_c$. Where the same rules apply as above. $W_c$ is not generally unique.
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**Center Manifold Theorem**:
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Let $f \in C^1(E), r\leq1$ where $E$ is an open subspace of R^n .
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If f(0)=0 and J has n_s eigenvalues with negative real part, n_u eigenvalues with positive real part, and if n_c = n-n_s-n_u purely imaginary eigenvalues exist,
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Then there exists an n_c dimensional center manifold $W_c$ of a class $C^r$ which is tangent to $E_c$.
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Examples in class slides.
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