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dynamical-system-functor skill

/skills/dynamical-system-functor

This skill analyzes and applies the dynamical system functor to reason about phase space evolution and stability under parameter changes.

npx playbooks add skill plurigrid/asi --skill dynamical-system-functor

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: dynamical-system-functor
description: Categorical structure of dynamical systems
trit: 1
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Dynamical System Functor

**Trit**: 1 (PLUS)
**Domain**: Dynamical Systems Theory
**Principle**: Categorical structure of dynamical systems

## Overview

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

## Mathematical Definition

```
DYNAMICAL_SYSTEM_FUNCTOR: 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

# Dynamical System Functor 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**: dynamical-system-functor
**Type**: Dynamical Systems / Dynamical System Functor
**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 encodes the categorical structure of dynamical systems as a compositional functor from phase space and time back to phase space. It captures local and global qualitative behavior of flows, equilibria, bifurcations, and stability inside a compositional algebraic framework. The skill is framed for use in compositional analysis and resource-sharing dynamical constructions.

How this skill works

The functor maps (phase space, time) pairs to updated phase states, enabling composition of dynamical components and preservation of structural invariants. It inspects local behavior near equilibria and invariant sets, computes global long-term structures like limit sets and attractors, and tracks bifurcation-driven qualitative changes. Algebraic composition rules and Möbius-style cancellation guide non-backtracking path composition and triadic conservation constraints.

When to use it

  • Modeling compositional dynamical systems where components share resources or states
  • Analyzing stability, attractors, and long-term behavior in a categorical framework
  • Studying parameter-driven bifurcations and qualitative transitions
  • Constructing non-backtracking skill chains with conserved trit sums
  • Integrating with algebraic dynamics tooling for compositional simulation

Best practices

  • Represent phase spaces and time parameters explicitly to preserve functorial composition
  • Isolate local equilibria and invariant sets before composing global behaviors
  • Use Möbius inversion or parity filters to cancel backtracking compositions
  • Validate GF(3) trit conservation across triplets when chaining skills
  • Check spectral gap conditions and expansion bounds when working with graph-based traversals

Example use cases

  • Compose several subsystem dynamics into a single resource-sharing machine and analyze emergent attractors
  • Detect and classify bifurcations as parameters vary in a compositional pipeline
  • Prove robustness of an equilibrium under perturbations using categorical stability arguments
  • Design non-backtracking traversal chains where composite path cancellation is enforced by Möbius filters
  • Integrate with AlgebraicDynamics-style libraries to implement oapply-like composition primitives

FAQ

What does the trit value mean?

The trit labels compositional role (here plus/1) used to ensure triadic conservation rules when skills are combined.

How does non-backtracking composition work?

Paths that revisit states are cancelled using Möbius-style inversion; only squarefree (non-backtracking) path lengths survive.