coupled-system

# Coupled System **Trit**: -1 (MINUS) **Domain**: Dynamical Systems Theory **Principle**: Interacting dynamical systems ## Overview Coupled System is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds. ## Mathematical Definition ``` COUPLED_SYSTEM: 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** (MINUS): Sinks/absorbers - **Conservation**: Σ trits ≡ 0 (mod 3) across skill triplets ## AlgebraicDynamics.jl Connection ```julia using AlgebraicDynamics # Coupled System 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**: coupled-system **Type**: Dynamical Systems / Coupled System **Trit**: -1 (MINUS) **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) ```