pitchfork
Symmetric bifurcation with symmetry breaking
When & Why to Use This Skill
This Claude skill provides a specialized framework for analyzing Pitchfork bifurcations within the domain of dynamical systems theory. It enables the modeling of symmetric bifurcations and symmetry breaking, allowing researchers to understand qualitative changes in differential equations, evaluate local and global stability, and explore the long-term dynamics of flows on manifolds. With integration capabilities for AlgebraicDynamics.jl, it bridges the gap between theoretical mathematical definitions and computational implementation.
Use Cases
- Stability Analysis: Evaluating the robustness of equilibria and invariant sets in physical or biological models under parameter perturbations.
- Computational Modeling: Implementing compositional dynamical systems using Julia's AlgebraicDynamics.jl for resource-sharing machine simulations.
- Theoretical Research: Investigating symmetry breaking and phase transitions in mathematical models of flows on manifolds and phase space.
- System Engineering: Analyzing qualitative behavior changes in complex systems to identify and manage critical tipping points and bifurcations.
| name | pitchfork |
|---|---|
| description | Symmetric bifurcation with symmetry breaking |
| version | 1.0.0 |
Pitchfork
Trit: 1 (PLUS) Domain: Dynamical Systems Theory Principle: Symmetric bifurcation with symmetry breaking
Overview
Pitchfork is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.
Mathematical Definition
PITCHFORK: Phase space × Time → Phase space
Key Properties
- Local behavior: Analysis near equilibria and invariant sets
- Global structure: Long-term dynamics and limit sets
- Bifurcations: Parameter-dependent qualitative changes
- 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
# Pitchfork 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: pitchfork Type: Dynamical Systems / Pitchfork 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:
- Prime Path: No state revisited in skill invocation chain
- Möbius Filter: Composite paths (backtracking) cancel via μ-inversion
- GF(3) Conservation: Trit sum ≡ 0 (mod 3) across skill triplets
- 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)