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The **Koopman operator** K linearizes nonlinear dynamics by lifting to infinite-dimensional observable space:
Provides local approximation of nonlinear dynamics for analyzing behavior near equilibria, stability, bifurcations, and long-term structure in dynamical systems.
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The Koopman operator K linearizes nonlinear dynamics by lifting to infinite-dimensional observable space:
State space (nonlinear) Observable space (linear)
x_{t+1} = f(x_t) → (Kg)(x) = g(f(x))
Key property: K is linear even when f is nonlinear.
DMD finds finite-rank approximation of K:
K ≈ Φ Λ Φ†
Koopman = natural transformation on observable presheaves:
# Observable functor
F: StateSpace → ObservableSpace
# Koopman as pushforward
K = f_*: Sh(X) → Sh(X)
dmd-spectral (-1) ⊗ structured-decomp (0) ⊗ koopman-generator (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ acsets (0) ⊗ koopman-generator (+1) = 0 ✓
This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:
dynamical-systems: 41 citations in bib.duckdbThis skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:
Trit: 1 (PLUS)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #4ECDC4
The skill participates in triads satisfying:
(-1) + (0) + (+1) ≡ 0 (mod 3)
This ensures compositional coherence in the Cat# equipment structure.