From b3f7fb125937f94cdb8646cc2f7aa5920ec21c72 Mon Sep 17 00:00:00 2001
From: Dane Sabo <dane.sabo@pitt.edu>
Date: Mon, 11 Nov 2024 13:24:57 -0500
Subject: [PATCH] vault backup: 2024-11-11 13:24:57

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+**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.
+**Case 2:** If the point is NOT hyperbolic. We've got to do something else.
+
+Assume $\vec{P} \in R^3$ is  a critical point in our system $\dot{X} = F(x), x\in R^3$
+Define stable and unstable manifolds of that point P as:
+$$ W_s(\vec{P}) = \left\{x: \Lambda^+(x) = \vec{P} \right\}$$
+$$ W_u(\vec{P}) = \left\{x: \Lambda^-(x) = \vec{P} \right\}$$
+Where the first is forward in time, the second is backward in time. 
+
+**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)
+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 $$\dot{x} = Jx$$
+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. 
+
+What they hell do we do when eigenvalues do not have a real part?
+**Center Manifold:** $W_c$ and **Center Eigenspace:** $E_c$. Where the same rules apply as above. $W_c$ is not generally unique.
+
+**Center Manifold Theorem**:
+Let $f \in C^1(E), r\leq1$ where $E$ is an open subspace of R^n .
+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,
+Then there exists an n_c dimensional center manifold $W_c$ of a class $C^r$ which is tangent to $E_c$. 
+
+Examples in class slides.