lasalle-invariance

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Invariance principle for asymptotic stability

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[Academic Research]

When & Why to Use This Skill

This Claude skill provides advanced mathematical tools for analyzing the asymptotic stability of dynamical systems using LaSalle's Invariance Principle. It enables researchers and engineers to determine the long-term qualitative behavior of differential equations and flows on manifolds, specifically addressing scenarios where traditional Lyapunov functions alone cannot guarantee convergence to an equilibrium point.

Use Cases

  • Control Theory: Proving the global asymptotic stability of non-linear systems where the Lyapunov derivative is only negative semi-definite.
  • Robotics and Automation: Analyzing the convergence of trajectories in complex mechanical systems to ensure they reach desired setpoints.
  • Agentic Workflow Optimization: Utilizing non-backtracking geodesic qualification to ensure efficient, loop-free execution of AI agent skill chains.
  • Mathematical Modeling: Investigating limit sets, bifurcations, and invariant sets in biological or chemical reaction networks.
namelasalle-invariance
descriptionInvariance principle for asymptotic stability
version1.0.0

LaSalle Invariance

Trit: 1 (PLUS) Domain: Dynamical Systems Theory Principle: Invariance principle for asymptotic stability

Overview

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

Mathematical Definition

LASALLE_INVARIANCE: 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

# LaSalle Invariance as compositional dynamical system
# Implements oapply for resource-sharing machines
  • equilibrium (trit 0)
  • stability (trit +1)
  • bifurcation (trit +1)
  • attractor (trit +1)
  • lyapunov-function (trit -1)

Skill Name: lasalle-invariance Type: Dynamical Systems / LaSalle Invariance 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)