**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.

Here's a more structured version of your notes, which could help with readability:

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# Nonlinear Dynamics: Manifolds and Critical Points

### Case 1: Hyperbolic Critical Point
If the critical point is **hyperbolic**, we can proceed with linearization:
- Linearize around the critical point.
- Analyze **eigenvalues** and **eigenvectors** to identify different **manifolds**.

### Case 2: Non-Hyperbolic Critical Point
If the critical point is **non-hyperbolic**, further techniques are required.

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Assume a critical point $\vec{P} \in \mathbb{R}^3$ for the system:  
$$ \dot{X} = F(x), \quad x \in \mathbb{R}^3 $$

Define the **stable** and **unstable manifolds** of point $\vec{P}$ as:
$$ W_s(\vec{P}) = \left\{ x : \lim_{t \to +\infty} \phi(t, x) = \vec{P} \right\} $$
$$ W_u(\vec{P}) = \left\{ x : \lim_{t \to -\infty} \phi(t, x) = \vec{P} \right\} $$
- \( W_s \): **Stable Manifold** (forward in time).
- \( W_u \): **Unstable Manifold** (backward in time).

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### Theorem: Existence of Stable and Unstable Manifolds
Given:
- \( x \) is a differential equation system in \( \mathbb{R}^n \).
- \( f \in C^1(E) \), with \( E \) an open subset of \( \mathbb{R}^n \) containing the origin.
- \( f(0) = 0 \) and the Jacobian \( J \) has \( n \) eigenvalues with non-zero real parts (**Hyperbolic**).

Then:
- In a small neighborhood around \( x \approx 0 \), stable and unstable manifolds \( W_s \) and \( W_u \) of the linearized system exist:
  $$ \dot{x} = Jx $$
- **Tangency Condition**: \( W_s \) and \( W_u \) are tangent to the eigenspaces \( E_s \) and \( E_u \) at \( x = 0 \).

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### Non-Real Eigenvalues
When eigenvalues do not have a real part:

- Define the **Center Manifold** \( W_c \) and **Center Eigenspace** \( E_c \).
- **Note**: \( W_c \) is generally **not unique**.

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### Center Manifold Theorem
Let:
- \( f \in C^1(E) \), \( r \leq 1 \), where \( E \) is an open subspace of \( \mathbb{R}^n \).
- \( f(0) = 0 \), and \( J \) (the Jacobian) has:
  - \( n_s \) eigenvalues with a negative real part.
  - \( n_u \) eigenvalues with a positive real part.
  - \( n_c = n - n_s - n_u \) purely imaginary eigenvalues.

Then there exists an \( n_c \)-dimensional **Center Manifold** \( W_c \) of class \( C^r \), which is tangent to \( E_c \).

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**Note**: Refer to class slides for detailed examples.