Birkhoff Average

Trit: 1 (PLUS) Domain: Dynamical Systems Theory Principle: Time average of observable along trajectory

Overview

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

Mathematical Definition

BIRKHOFF_AVERAGE: Phase space × Time → Phase space

Key Properties

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

Integration with GF(3)

This skill participates in triadic composition:

• Trit 1 (PLUS): Sources/generators
• Conservation: Σ trits ≡ 0 (mod 3) across skill triplets

AlgebraicDynamics.jl Connection

using AlgebraicDynamics

# Birkhoff Average as compositional dynamical system
# Implements oapply for resource-sharing machines

Related Skills

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

Skill Name: birkhoff-average Type: Dynamical Systems / Birkhoff Average Trit: 1 (PLUS) GF(3): Conserved in triplet composition

Non-Backtracking Geodesic Qualification

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

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)

SDF Interleaving

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

birkhoff-average (○) + SDF.Ch8 (−) + [balancer] (+) = 0

Skill Trit: 0 (ERGODIC – coordination)

Secondary Chapters

• Ch3: Variations on an Arithmetic Theme

Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.