Manifolds and Critical Points

Case 1: Hyperbolic Critical Point

If the critical point is hyperbolic, we can proceed with linearization:

Linearize around the critical point.
Analyze eigenvalues and eigenvectors to identify different manifolds.

Case 2: Non-Hyperbolic Critical Point

If the critical point is non-hyperbolic, further techniques are required.

Assume a critical point \vec{P} \in \mathbb{R}^3 for the system:

\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).

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).

Then:

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.

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.

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 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:

Stable Nodes
Stable Limit Cycles
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.

2.8 KiB Raw Permalink Blame History