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transcritical skill

This skill analyzes transcritical bifurcation in dynamical systems, helping you understand stability exchange between equilibria and predict qualitative

npx playbooks add skill plurigrid/asi --skill transcritical

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: transcritical
description: Bifurcation exchanging stability between equilibria
version: 1.0.0
---


# Transcritical

**Trit**: 0 (ERGODIC)
**Domain**: Dynamical Systems Theory
**Principle**: Bifurcation exchanging stability between equilibria

## Overview

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

## Mathematical Definition

```
TRANSCRITICAL: 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

# Transcritical 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**: transcritical
**Type**: Dynamical Systems / Transcritical
**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)
```

## 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

```
transcritical (○) + 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 models the transcritical bifurcation, a basic mechanism where stability is exchanged between equilibria as a parameter changes. It provides a compact, topologically minded toolkit to identify, analyze, and reason about local and global changes in flow caused by transcritical interactions.

How this skill works

The skill inspects vector fields or parameterized differential equations to detect equilibria collisions and stability exchanges. It computes local linearizations, tracks eigenvalue crossings, and reports normal-form diagnostics that characterize the transcritical unfolding. It also integrates with compositional frameworks to maintain triadic conservation constraints and to interleave robustness strategies.

When to use it

  • You need to determine how equilibria swap stability when a parameter varies.
  • Analyzing local bifurcations near intersecting fixed-point branches.
  • Building reduced normal-form models for parameter studies or continuation.
  • Composing dynamical behaviors in modular systems that enforce conservation constraints.
  • Designing control or fallback strategies that depend on qualitative regime changes.

Best practices

  • Linearize around candidate equilibria and verify nondegeneracy conditions before trusting normal-form output.
  • Combine local analysis with trajectory sampling to expose global consequences and basin changes.
  • Use parameter continuation tools to follow branches and detect exchange points robustly.
  • Maintain modular composition with conserved trit bookkeeping when combining multiple skills.
  • Validate predictions with small perturbation experiments to check structural stability.

Example use cases

  • Determine the critical parameter where two fixed branches exchange stability in a chemical reaction model.
  • Produce a normal-form reduction for a two-parameter study in ecology to explain species replacement.
  • Integrate with a compositional dynamical pipeline that enforces GF(3)-like conservation across components.
  • Design controller switching logic that anticipates regime shifts caused by a transcritical event.
  • Teach bifurcation concepts with a clear example showing stability exchange and phase portrait changes.

FAQ

What is the minimal diagnostic to confirm a transcritical bifurcation?

Check for two equilibria that collide at a parameter value, verify a simple eigenvalue crossing through zero, and confirm nondegeneracy coefficients in the normal form.

Does this skill handle global attractor changes?

It focuses on local detection and normal-form characterization; combine it with trajectory sampling or global analysis modules to assess long-term attractor structure.