lyapunov-stability
Stability via Lyapunov's direct method
When & Why to Use This Skill
This Claude skill provides advanced capabilities for analyzing the stability of dynamical systems using Lyapunov's direct method. It facilitates the qualitative assessment of differential equations, helping users identify equilibria, invariant sets, and long-term system behavior without requiring explicit solutions. By integrating with frameworks like AlgebraicDynamics.jl, it supports complex system modeling, bifurcation analysis, and robustness testing, making it an essential tool for researchers in mathematics, physics, and control engineering.
Use Cases
- Control Engineering: Proving the global asymptotic stability of non-linear control systems in robotics, aerospace, and automated machinery.
- Theoretical Physics: Analyzing phase space trajectories and limit sets to understand the qualitative behavior of physical flows and manifolds.
- Mathematical Research: Investigating bifurcations and parameter-dependent changes in dynamical systems to predict regime shifts and stability boundaries.
- Optimization & AI: Applying Lyapunov functions to guarantee the convergence and stability of iterative optimization algorithms and neural network training dynamics.
| name | lyapunov-stability |
|---|---|
| description | Stability via Lyapunov's direct method |
| version | 1.0.0 |
Lyapunov Stability
Trit: 0 (ERGODIC) Domain: Dynamical Systems Theory Principle: Stability via Lyapunov's direct method
Overview
Lyapunov Stability is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.
Mathematical Definition
LYAPUNOV_STABILITY: 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
# Lyapunov Stability 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: lyapunov-stability Type: Dynamical Systems / Lyapunov Stability 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)