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eigenvalue-stability skill

/skills/eigenvalue-stability

This skill analyzes system stability by Jacobian eigenvalues to help you classify equilibria and anticipate bifurcations.

npx playbooks add skill plurigrid/asi --skill eigenvalue-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: eigenvalue-stability
description: Stability classification via Jacobian eigenvalues
version: 1.0.0
---


# Eigenvalue Stability

**Trit**: 1 (PLUS)
**Domain**: Dynamical Systems Theory
**Principle**: Stability classification via Jacobian eigenvalues

## Overview

Eigenvalue 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

```
EIGENVALUE_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

# Eigenvalue 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**: eigenvalue-stability
**Type**: Dynamical Systems / Eigenvalue 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

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

**Skill Trit**: 0 (ERGODIC - coordination)

### Secondary Chapters

- Ch3: Variations on an Arithmetic Theme

### Connection Pattern

Degeneracy provides fallbacks. This skill offers redundant strategies.

Overview

This skill implements stability classification for dynamical systems by inspecting Jacobian eigenvalues around equilibria and invariant sets. It gives a practical, local criterion for asymptotic, marginal, and unstable behavior and ties that classification to bifurcation and long-term structure. The skill is lightweight, focused on eigenvalue-based diagnostics and integration into compositional analyses.

How this skill works

Given a vector field and a candidate equilibrium, the skill computes the Jacobian matrix and its spectrum, then classifies stability from eigenvalue signs and real parts. It flags simple bifurcations (saddle-node, Hopf) when eigenvalues cross critical boundaries and reports spectral gaps relevant to robustness. Outputs include classification labels, leading eigenvectors, and numeric indicators for perturbation sensitivity.

When to use it

  • Classifying local stability of equilibria in ODE models
  • Detecting parameter values where qualitative behavior changes (bifurcation pre-check)
  • Screening candidate steady states for robustness under small perturbations
  • Integrating with compositional analyses that require trit-based conservation or spectral annotations
  • Evaluating whether a system has spectral gaps that imply rapid mixing or slow manifolds

Best practices

  • Evaluate Jacobian at precisely computed equilibria; avoid noisy numerical roots
  • Use complex eigenvalue analysis to detect oscillatory instabilities (Hopf)
  • Report both eigenvalues and right/left eigenvectors for mode interpretation
  • Complement eigenvalue checks with Lyapunov or numerical simulation for non-hyperbolic cases
  • Check condition numbers and regularize Jacobian if near-singular to avoid misclassification

Example use cases

  • Automatically classifying stability of metabolic network steady states during model calibration
  • Pre-filtering parameter sets in bifurcation studies to focus continuation on likely transitions
  • Annotating nodes in a compositional workflow where trit conservation informs orchestration
  • Detecting emergent oscillations by identifying complex-conjugate eigenpairs crossing the imaginary axis
  • Identifying dominant modes for model reduction by ranking eigenvalues and spectral gaps

FAQ

What does a complex eigenvalue with positive real part mean?

It indicates an unstable oscillatory mode; trajectories spiral away from the equilibrium.

How do you handle zero or near-zero eigenvalues?

Treat them as non-hyperbolic: augment with higher-order analysis, Lyapunov methods, or numerical continuation.