lyapunov-function

Scalar function decreasing along trajectories

16 stars

byplurigrid

View on GitHub Installation ↓

Best use case

lyapunov-function is best used when you need a repeatable AI agent workflow instead of a one-off prompt.

Scalar function decreasing along trajectories

Teams using lyapunov-function should expect a more consistent output, faster repeated execution, less prompt rewriting.

When to use this skill

You want a reusable workflow that can be run more than once with consistent structure.

When not to use this skill

You only need a quick one-off answer and do not need a reusable workflow.
You cannot install or maintain the underlying files, dependencies, or repository context.

Installation

Claude Code / Cursor / Codex

$curl -o ~/.claude/skills/lyapunov-function/SKILL.md --create-dirs "https://raw.githubusercontent.com/plurigrid/asi/main/plugins/asi/skills/lyapunov-function/SKILL.md"

Manual Installation

Download SKILL.md from GitHub
Place it in .claude/skills/lyapunov-function/SKILL.md inside your project
Restart your AI agent — it will auto-discover the skill

How lyapunov-function Compares

Feature / Agent	lyapunov-function	Standard Approach
Platform Support	Not specified	Limited / Varies
Context Awareness	High	Baseline
Installation Complexity	Unknown	N/A

Frequently Asked Questions

What does this skill do?

Scalar function decreasing along trajectories

Where can I find the source code?

You can find the source code on GitHub using the link provided at the top of the page.

SKILL.md Source

# Lyapunov Function

**Trit**: -1 (MINUS)
**Domain**: Dynamical Systems Theory
**Principle**: Scalar function decreasing along trajectories

## Overview

Lyapunov Function 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_FUNCTION: 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

```julia
using AlgebraicDynamics

# Lyapunov Function 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-function
**Type**: Dynamical Systems / Lyapunov Function
**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)
```

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