75 lines
2.8 KiB
Markdown
75 lines
2.8 KiB
Markdown
# Manifolds and Critical Points
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### Case 1: Hyperbolic Critical Point
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If the critical point is **hyperbolic**, we can proceed with linearization:
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- Linearize around the critical point.
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- Analyze **eigenvalues** and **eigenvectors** to identify different **manifolds**.
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### Case 2: Non-Hyperbolic Critical Point
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If the critical point is **non-hyperbolic**, further techniques are required.
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---
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Assume a critical point $\vec{P} \in \mathbb{R}^3$ for the system:
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$$ \dot{X} = F(x), \quad x \in \mathbb{R}^3 $$
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Define the **stable** and **unstable manifolds** of point $\vec{P}$ as:
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$$ W_s(\vec{P}) = \left\{ x : \lim_{t \to +\infty} \phi(t, x) = \vec{P} \right\} $$
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$$ W_u(\vec{P}) = \left\{ x : \lim_{t \to -\infty} \phi(t, x) = \vec{P} \right\} $$
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- $W_s$: **Stable Manifold** (forward in time).
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- $W_u$: **Unstable Manifold** (backward in time).
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### Theorem: Existence of Stable and Unstable Manifolds
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Given:
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- $x$ is a differential equation system in $\mathbb{R}^n$.
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- $f \in C^1(E)$, with $E$ an open subset of $\mathbb{R}^n$ containing the origin.
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- $f(0) = 0$ and the Jacobian $J$ has $n$ eigenvalues with non-zero real parts (**Hyperbolic**).
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Then:
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- In a small neighborhood around $x \approx 0$, stable and unstable manifolds $W_s$ and $W_u$ of the linearized system exist:
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$$ \dot{x} = Jx $$
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- **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
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When eigenvalues do not have a real part:
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- Define the **Center Manifold** $W_c$ and **Center Eigenspace** $E_c$.
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- **Note**: $W_c$ is generally **not unique**.
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### Center Manifold Theorem
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Let:
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- $f \in C^1(E)$, $r \leq 1$, where $E$ is an open subspace of $\mathbb{R}^n$.
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- $f(0) = 0$, and $J$ (the Jacobian) has:
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- $n_s$ eigenvalues with a negative real part.
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- $n_u$ eigenvalues with a positive real part.
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- $n_c = n - n_s - n_u$ purelyDeterministic Chaos = imaginary eigenvalues.
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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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---
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**Note**: Refer to class slides for detailed examples.
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# Attractors
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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.
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There are four types of attractors:
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1. Stable Nodes
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2. Stable Limit Cycles
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3. Strange Atractor (3D)
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1. Coined by Otto Roessler (1976)
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Here's an example:
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$$ \dot x = -(y+z)$$
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$$ \dot y = x+ay$$
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$$ \dot z = b + xz - cz$$
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$c=6.3$, $a, b = 0.2$
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Behavior appears random but comes from simple deterministic equations
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**Deterministic Chaos** Arises from determinsitic state equations and ICS
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**Nondeterministic Chaos** no underlying equations, or noisy, and random input.
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We care more about deterministic chaos.
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Bajaj then shows a code he made. The Roessler attractor. |