time-parameterization

Time Parameterization is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.

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TIME_PARAMETERIZATION: Phase space × Time → Phase space

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1. Local behavior: Analysis near equilibria and invariant sets
2. Global structure: Long-term dynamics and limit sets
3. Bifurcations: Parameter-dependent qualitative changes
4. Stability: Robustness under perturbation

This skill participates in triadic composition:

• Trit -1 (MINUS): Sinks/absorbers
• Conservation: Σ trits ≡ 0 (mod 3) across skill triplets

```julia

using AlgebraicDynamics

# Time Parameterization as compositional dynamical system

# Implements oapply for resource-sharing machines

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• equilibrium (trit 0)
• stability (trit +1)
• bifurcation (trit +1)
• attractor (trit +1)
• lyapunov-function (trit -1)

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Skill Name: time-parameterization

Type: Dynamical Systems / Time Parameterization

Trit: -1 (MINUS)

GF(3): Conserved in triplet composition

Condition: μ(n) ≠ 0 (Möbius squarefree)

This skill is qualified for non-backtracking geodesic traversal:

1. Prime Path: No state revisited in skill invocation chain
2. Möbius Filter: Composite paths (backtracking) cancel via μ-inversion
3. GF(3) Conservation: Trit sum ≡ 0 (mod 3) across skill triplets
4. Spectral Gap: Ramanujan bound λ₂ ≤ 2√(k-1) for k-regular expansion

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Geodesic Invariant:

∀ path P: backtrack(P) = ∅ ⟹ μ(|P|) ≠ 0

Möbius Inversion:

f(n) = Σ_{d|n} g(d) ⟹ g(n) = Σ_{d|n} μ(n/d) f(d)

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This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 8. Degeneracy

Concepts: redundancy, fallback, multiple strategies, robustness

GF(3) Balanced Triad

```

time-parameterization (+) + SDF.Ch8 (−) + [balancer] (○) = 0

```

Skill Trit: 1 (PLUS - generation)

Secondary Chapters

• Ch3: Variations on an Arithmetic Theme

Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.