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stable-manifold skill

/skills/stable-manifold

This skill analyzes the stable manifold of dynamical systems to describe convergence to equilibria and informs long-term behavior and stability.

npx playbooks add skill plurigrid/asi --skill stable-manifold

Review the files below or copy the command above to add this skill to your agents.

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SKILL.md
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---
name: stable-manifold
description: Manifold of points converging to equilibrium
trit: 0
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Stable Manifold

**Trit**: 0 (ERGODIC)
**Domain**: Dynamical Systems Theory
**Principle**: Manifold of points converging to equilibrium

## Overview

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

## Mathematical Definition

```
STABLE_MANIFOLD: 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 0** (ERGODIC): Neutral/ergodic
- **Conservation**: Σ trits ≡ 0 (mod 3) across skill triplets

## AlgebraicDynamics.jl Connection

```julia
using AlgebraicDynamics

# Stable Manifold 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**: stable-manifold
**Type**: Dynamical Systems / Stable Manifold
**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:

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)
```

Overview

This skill describes the stable manifold: the set of initial conditions whose trajectories converge to an equilibrium under a flow or map. It frames local and global behavior around invariant sets and links stability analysis with compositional constraints in a triadic GF(3) framework. The presentation emphasizes practical checks for convergence, bifurcation sensitivity, and non-backtracking traversal qualifications.

How this skill works

The skill inspects phase space structure near equilibria and computes or characterizes the manifold of points that asymptotically approach those equilibria. It highlights local linearization, invariant manifold theorems, and global limit-set relationships. It also encodes a modular composition rule (trit 0) used when combining skills in triplets, and provides non-backtracking/geodesic qualifications using Möbius filtering and spectral-gap criteria.

When to use it

  • Analyzing long-term behavior of differential equations or discrete maps
  • Identifying basins of attraction for steady states or fixed points
  • Studying bifurcations that change convergence properties
  • Composing dynamical primitives while preserving triadic GF(3) constraints
  • Designing or validating non-backtracking traversal rules in state-space analyses

Best practices

  • Linearize system near equilibria to obtain local manifold approximation before global refinement
  • Use invariant manifold theorems to justify reduced models and dimensionality reduction
  • Check robustness under perturbations to assess structural stability
  • When composing skills, enforce trit conservation (sum ≡ 0 mod 3) to maintain ergodic neutrality
  • Apply Möbius-filtered non-backtracking tests and spectral-gap bounds to validate path-based analyses

Example use cases

  • Compute the local stable manifold for a saddle or sink to determine the basin boundary
  • Reduce a high-dimensional model by restricting dynamics to the stable manifold near an attractor
  • Detect how parameter changes produce bifurcations that alter convergence and basin geometry
  • Compose stability-related skills (stability, bifurcation, attractor) while maintaining triadic conservation
  • Apply non-backtracking geodesic qualification to path-sampling in networked dynamical models

FAQ

What does trit 0 (ERGODIC) mean here?

Trit 0 signals a neutral or ergodic role in compositional rules; when combined in triplets it preserves a GF(3) conservation law (sum ≡ 0 mod 3).

How does the Möbius filter apply to paths?

Composite or backtracking paths cancel via Möbius inversion: only squarefree path lengths (μ(n) ≠ 0) survive the filter, enforcing a non-backtracking prime-path criterion.