Obsidian/300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-11-11 Nonlinear 3D Phenomena.md

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:

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Nonlinear Dynamics: 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.

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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).

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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.

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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.

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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.

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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Note: Refer to class slides for detailed examples.