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

@@ -1,30 +1,3 @@
-**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:
-
----
-
 # Nonlinear Dynamics: Manifolds and Critical Points
 
 ### Case 1: Hyperbolic Critical Point
@@ -43,42 +16,42 @@ $$ \dot{X} = F(x), \quad x \in \mathbb{R}^3 $$
 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).
+- $W_s$: **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**).
+- $x$ 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:
+- 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 \( x = 0 \).
+- **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**.
+- 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.
+- $f \in C^1(E)$, $r \leq 1$, 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 \( E_c \).
+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.
+**Note**: Refer to class slides for detailed examples.