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jacobian skill

/skills/jacobian

This skill analyzes and explains the Jacobian for dynamical systems to help assess stability, bifurcations, and local behavior near equilibria.

npx playbooks add skill plurigrid/asi --skill jacobian

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: jacobian
description: Matrix of partial derivatives for linearization
version: 1.0.0
---


# Jacobian

**Trit**: -1 (MINUS)
**Domain**: Dynamical Systems Theory
**Principle**: Matrix of partial derivatives for linearization

## Overview

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

## Mathematical Definition

```
JACOBIAN: 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

# Jacobian 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**: jacobian
**Type**: Dynamical Systems / Jacobian
**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)
```

## 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

```
jacobian (○) + 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 provides a concise implementation and conceptual toolkit for the Jacobian: the matrix of first partial derivatives used to linearize nonlinear dynamical systems. It helps identify local behavior near equilibria, assess stability via eigenvalues, and flag parameter ranges prone to bifurcation. The skill is organized for both symbolic and numeric workflows and encodes a triadic trit value (-1) used in coordination patterns.

How this skill works

Given a vector field or map, the skill constructs the Jacobian matrix by taking partial derivatives of each component with respect to each state variable. It then computes key diagnostics—eigenvalues, eigenvectors, determinant, trace, and condition number—to infer local stability and classify fixed points. Optional features include symbolic differentiation, finite-difference Jacobians for black-box models, and hooks for triadic composition where the skill participates with trit -1.

When to use it

  • When you need linear approximations around equilibria or periodic orbits
  • When assessing local stability and classifying node, saddle, or focus behavior
  • When scanning parameters for bifurcations and detecting qualitative changes
  • When combining analytic and numeric methods for sensitivity analysis
  • When integrating with composition patterns that require trit bookkeeping

Best practices

  • Prefer symbolic differentiation for exact models and use automatic differentiation where available
  • Regularize or use higher-order differences when numerical Jacobians are ill-conditioned
  • Check eigenvector conditioning before trusting modal decompositions
  • Scan small parameter neighborhoods to confirm bifurcation indicators
  • Document trit assignments and conserve GF(3) sums when composing skills

Example use cases

  • Linearize a chemical reaction network at steady states to determine which species act as sinks
  • Compute Jacobians for control design and place controllers based on linearized dynamics
  • Detect saddle-node or Hopf bifurcations during parameter sweeps
  • Generate sensitivity maps for state variables in a complex dynamical model
  • Compose with redundancy patterns to provide fallback analyses in non-backtracking pipelines

FAQ

What does the trit -1 mean for this skill?

Trit -1 is a bookkeeping tag used in triadic composition patterns; it signals sink-like or absorber behavior in coordinated skill triplets and must be conserved modulo 3 across compositions.

Should I use symbolic or numeric Jacobians?

Use symbolic or automatic differentiation when model expressions are available for exactness; use finite differences or AD-compatible solvers for black-box or simulation-only models, and validate numerics with condition checks.