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stability skill

/skills/stability

This skill helps you understand qualitative stability of dynamical systems near equilibria, guiding robust analysis of perturbations and long-term behavior.

npx playbooks add skill plurigrid/asi --skill stability

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: stability
description: Qualitative behavior of solutions near equilibria
version: 1.0.0
---


# Stability

**Trit**: 1 (PLUS)
**Domain**: Dynamical Systems Theory
**Principle**: Qualitative behavior of solutions near equilibria

## Overview

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

```
STABILITY: 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

```julia
using AlgebraicDynamics

# 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**: stability
**Type**: Dynamical Systems / Stability
**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)
```

## SDF Interleaving

This skill connects to **Software Design for Flexibility** (Hanson & Sussman, 2021):

### Primary Chapter: 8. Degeneracy

**Concepts**: redundancy, fallback, multiple strategies, robustness

### GF(3) Balanced Triad

```
stability (−) + SDF.Ch8 (−) + [balancer] (−) = 0
```

**Skill Trit**: -1 (MINUS - verification)

### Secondary Chapters

- Ch3: Variations on an Arithmetic Theme

### Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.

Overview

This skill describes qualitative behavior of dynamical systems near equilibria and invariant sets. It frames local and global stability, robustness under perturbations, and how parameter changes lead to bifurcations. The skill is oriented to practitioners who need conceptual tools for analyzing long-term behavior and designing resilient system responses.

How this skill works

The skill inspects phase space structure and time evolution to classify equilibrium types, attractors, and limit sets. It uses local linearization, Lyapunov ideas, and topological invariants to assess stability and robustness. It also highlights how parameters induce bifurcations and how triadic composition rules (GF(3) trits) track source/verification roles in composed analyses.

When to use it

  • Diagnosing whether an equilibrium is stable, asymptotically stable, or unstable
  • Evaluating robustness of system behavior under small perturbations or model uncertainty
  • Exploring how changing parameters produce bifurcations and qualitative regime shifts
  • Designing controllers or fallback strategies that rely on attractors and invariant manifolds
  • Composing multiple analysis modules while preserving role balance in triadic compositions (GF(3) bookkeeping)

Best practices

  • Start with local linearization around equilibria and verify conclusions against nonlinear terms
  • Construct or search for Lyapunov functions to certify stability when linearization is inconclusive
  • Analyze global limit sets and invariant manifolds to understand long-term behavior beyond local neighborhoods
  • When studying parameter dependence, map bifurcation diagrams and test representative parameter slices
  • Use redundant strategies (degeneracy) and explicit fallback controllers to increase robustness under model mismatch

Example use cases

  • Determine stability of fixed points in an ODE model of a chemical reaction network
  • Validate a candidate Lyapunov function numerically and symbolically for a control system
  • Map bifurcation structure of a population model to predict regime transitions
  • Design resilient routines that route control to backup strategies when primary attractors degrade
  • Compose analysis modules with trit accounting to ensure conservation of source/verification roles

FAQ

How does local linearization relate to true nonlinear stability?

Linearization gives first-order information: if eigenvalues have negative real parts the equilibrium is locally asymptotically stable; if any eigenvalue has positive real part it is unstable. Nonhyperbolic cases require Lyapunov methods or center-manifold reduction.

When should I seek a Lyapunov function?

Use Lyapunov functions when linearization is inconclusive or when you need global stability guarantees. Constructive or numerical candidates can often certify asymptotic stability and robustness to perturbations.