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Claude-skill-registry hopf

Bifurcation creating limit cycle from equilibrium

install
source · Clone the upstream repo
git clone https://github.com/majiayu000/claude-skill-registry
Claude Code · Install into ~/.claude/skills/
T=$(mktemp -d) && git clone --depth=1 https://github.com/majiayu000/claude-skill-registry "$T" && mkdir -p ~/.claude/skills && cp -r "$T/skills/data/hopf" ~/.claude/skills/majiayu000-claude-skill-registry-hopf && rm -rf "$T"
manifest: skills/data/hopf/SKILL.md
source content

Hopf

Trit: -1 (MINUS) Domain: Dynamical Systems Theory Principle: Bifurcation creating limit cycle from equilibrium

Overview

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

Mathematical Definition

HOPF: 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

using AlgebraicDynamics

# Hopf 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: hopf Type: Dynamical Systems / Hopf 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)

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

hopf (−) + SDF.Ch8 (−) + [balancer] (−) = 0

Skill Trit: -1 (MINUS - verification)

Secondary Chapters

  • Ch3: Variations on an Arithmetic Theme

Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.