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.

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 purely 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.

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