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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.