# 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.
  - $n_c = n - n_s - n_u$ purelyDeterministic Chaos = 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.

# Attractors
An attractor is a minimal, closed, invariant set that 'attracts' nearby trajectories lying in some domain of stability (or, in other words, a basin of attraction) onto it. 
There are four types of attractors:
1. Stable Nodes
2. Stable Limit Cycles
3. Strange Atractor (3D)
	1. Coined by Otto Roessler (1976)
Here's an example:
$$ \dot x = -(y+z)$$
$$ \dot y = x+ay$$
$$ \dot z = b + xz - cz$$
$c=6.3$, $a, b = 0.2$
Behavior appears random but comes from simple deterministic equations
**Deterministic Chaos** Arises from determinsitic state equations and ICS
**Nondeterministic Chaos** no underlying equations, or noisy, and random input.
We care more about deterministic chaos.
Bajaj then shows a code he made. The Roessler attractor.