home / skills / plurigrid / asi / saddle-node

saddle-node skill

safe

This skill helps analyze saddle-node bifurcations in dynamical systems, identifying equilibrium creation and destruction and guiding stability assessment.

npx playbooks add skill plurigrid/asi --skill saddle-node

Review the files below or copy the command above to add this skill to your agents.

Files (1)

SKILL.md

2.0 KB

---
name: saddle-node
description: Bifurcation creating/destroying equilibrium pair
trit: -1
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Saddle-node

**Trit**: -1 (MINUS)
**Domain**: Dynamical Systems Theory
**Principle**: Bifurcation creating/destroying equilibrium pair

## Overview

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

## Mathematical Definition

```
SADDLE-NODE: 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

# Saddle-node 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**: saddle-node
**Type**: Dynamical Systems / Saddle-node
**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)
```

Overview

This skill models the saddle-node bifurcation, the local creation or annihilation of an equilibrium pair in a continuous dynamical system. It captures the topological and qualitative consequences of that bifurcation for flows on manifolds and for parameter-dependent families. The representation emphasizes compositional use in modular dynamical pipelines and preserves a trit label of -1 for compositional bookkeeping.

How this skill works

The skill inspects local phase-space structure near an equilibrium and identifies conditions where a pair of fixed points (one stable, one unstable) is created or destroyed as a parameter passes a critical value. It provides diagnostics for local normal-form reduction, stability assignment, and how the bifurcation affects long-term limit sets. Integration hooks encode a GF(3) trit bookkeeping convention so the skill composes predictably with complementary skills.

When to use it

Analyzing parameter sweeps where equilibria appear or disappear
Reducing local dynamics to normal form to classify stability changes
Composing with attractor or stability skills to track qualitative shifts
Building modular dynamical models that require trit-based composition
Filtering non-backtracking geodesic traversals where prime-path qualification matters

Best practices

Perform local coordinate or center-manifold reduction before invoking the skill
Validate normal-form coefficients numerically for noisy or high-dimensional systems
Compose with explicit stability and equilibrium skills to maintain GF(3) conservation
Use Möbius filtering for path-based analyses to avoid backtracking artifacts
Record parameter values at bifurcation onset for reproducibility and continuation

Example use cases

Detecting saddle-node onset in a chemical reaction network during parameter continuation
Annotating a phase portrait with newly created sink/source pairs after a perturbation
Composing with a Lyapunov-function skill to assess robustness of the emergent equilibrium
Applying non-backtracking geodesic qualification to explore prime-path transitions in a skill graph
Using the skill as a building block in AlgebraicDynamics-style compositional models

FAQ

What does the trit -1 mean in practice?

Trit -1 marks this skill as a sink/absorber type in triadic composition so sums across triplets conserve mod 3 when combined with complementary skills.

When should I prefer saddle-node over a generic bifurcation check?

Use saddle-node when you expect a coalescence of two equilibria with typical nondegeneracy conditions; for Hopf or transcritical behavior choose the corresponding specialized skill.