From 7bde0e4fdc87e2500ef4658bdf6915b55afa44ab Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 11 Nov 2024 13:37:05 -0500 Subject: [PATCH] vault backup: 2024-11-11 13:37:05 --- .../2024-11-11 Nonlinear 3D Phenomena.md | 59 +++++-------------- 1 file changed, 16 insertions(+), 43 deletions(-) diff --git a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-11 Nonlinear 3D Phenomena.md b/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-11 Nonlinear 3D Phenomena.md index 2e16456b..f3b5d8b1 100644 --- a/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-11 Nonlinear 3D Phenomena.md +++ b/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. \ No newline at end of file +**Note**: Refer to class slides for detailed examples.