Skill

vector-field

From asi

Defines vector fields in dynamical systems theory, covering phase space dynamics, local/global behavior, stability, bifurcations, and Julia AlgebraicDynamics.jl integration. Useful for analyzing differential equations.

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**Trit**: 0 (ERGODIC)

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Last CommitFeb 16, 2026

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Vector Field

Trit: 0 (ERGODIC) Domain: Dynamical Systems Theory Principle: Assignment of vectors to points in phase space defining dynamics

Overview

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

Mathematical Definition

VECTOR_FIELD: 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 0 (ERGODIC): Neutral/ergodic
Conservation: Σ trits ≡ 0 (mod 3) across skill triplets

AlgebraicDynamics.jl Connection

using AlgebraicDynamics

# Vector Field 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: vector-field Type: Dynamical Systems / Vector Field Trit: 0 (ERGODIC) 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)