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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-11 Nonlinear 3D Phenomena.md

@@ -20,3 +20,65 @@ If f(0)=0 and J has n_s eigenvalues with negative real part, n_u eigenvalues wit
 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:
+
+---
+
+# 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.
+
+---
+
+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).
+
+---
+
+### 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 \( x = 0 \).
+
+---
+
+### 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**.
+
+---
+
+### 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 \( E_c \).
+
+---
+
+**Note**: Refer to class slides for detailed examples.