home / skills / plurigrid / asi / eigenvalue-stability

eigenvalue-stability skill

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This skill analyzes dynamical system stability by examining Jacobian eigenvalues to classify equilibria and predict long-term behavior.

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
trit: 1
geodesic: true
moebius: "μ(n) ≠ 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)
```

Overview

This skill classifies dynamical stability by inspecting the eigenvalues of Jacobian matrices at equilibria and invariant sets. It provides a concise, practical workflow for deciding local stability, detecting bifurcations, and interpreting spectral gaps in compositional systems. The presentation emphasizes outcomes useful for modeling, analysis, and automated pipelines.

How this skill works

The skill computes the Jacobian of a vector field at a point or along an invariant set, then evaluates its spectrum. Real parts of eigenvalues determine local attractivity or repulsion; zero-crossings and repeated eigenvalues indicate possible bifurcations. It can be composed into larger pipelines where trit conservation and spectral gap checks guide admissible transitions.

When to use it

Determining local stability of equilibria in ODE models.
Screening candidate steady states during parameter sweeps.
Detecting Hopf, saddle-node, or transcritical bifurcations via eigenvalue movement.
Assessing robustness of attractors under small perturbations.
Embedding stability checks into compositional dynamical workflows that require spectral invariants.

Best practices

Compute the full Jacobian analytically when possible; use high-precision numerics for near-degenerate spectra.
Inspect both eigenvalues and eigenvectors to identify center manifolds or neutral directions.
Combine eigenvalue analysis with Lyapunov functions or numerical trajectories for global confirmation.
Monitor parameter continuations to catch crossings through the imaginary axis early.
Validate spectral gap assumptions before invoking results that rely on expansion or Ramanujan-type bounds.

Example use cases

Classify equilibria of a chemical reaction network by linearizing reaction-rate ODEs and reading eigenvalue signs.
Automate bifurcation detection in parameter scans by tracking eigenvalue real-part sign changes.
Integrate stability checks into a compositional pipeline that enforces trit conservation across skill triplets.
Use spectral gap criteria to validate expansion properties in networked or discrete dynamical systems.
Filter candidate attractors during model discovery by rejecting states with unstable spectra.

FAQ

What does a complex-conjugate pair with positive real part imply?

It indicates a locally unstable focus; trajectories spiral away from the equilibrium and a Hopf bifurcation may occur if the real parts cross zero.

How do I handle zero or near-zero eigenvalues?

Zero eigenvalues signal center directions or bifurcation onset; use center-manifold reduction or higher-order analysis and increase numerical precision to resolve near-degenerate cases.