From 32bede56f56820d977d68f5568fe9d3cfa88fcfd Mon Sep 17 00:00:00 2001 From: Dane Sabo Date: Mon, 11 Nov 2024 14:05:41 -0500 Subject: [PATCH] vault backup: 2024-11-11 14:05:40 --- .../2024-11-11 Nonlinear 3D Phenomena.md | 22 +++++++++++++++++-- 1 file changed, 20 insertions(+), 2 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 f3b5d8b1..8daf7551 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 @@ -1,4 +1,4 @@ -# Nonlinear Dynamics: Manifolds and Critical Points +# Manifolds and Critical Points ### Case 1: Hyperbolic Critical Point If the critical point is **hyperbolic**, we can proceed with linearization: @@ -48,10 +48,28 @@ Let: - $f(0) = 0$, 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. + - $n_c = n - n_s - n_u$ purelyDeterministic Chaos = imaginary eigenvalues. Then there exists an $n_c$-dimensional **Center Manifold** $W_c$ of class $C^r$, which is tangent to $E_c$. --- **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. \ No newline at end of file