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repeller skill

/skills/repeller

This skill helps analyze and repel nearby trajectories in dynamical systems, aiding understanding of long-term behavior and stability near invariant sets.

npx playbooks add skill plurigrid/asi --skill repeller

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: repeller
description: Invariant set repelling nearby trajectories
version: 1.0.0
---


# Repeller

**Trit**: -1 (MINUS)
**Domain**: Dynamical Systems Theory
**Principle**: Invariant set repelling nearby trajectories

## Overview

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

## Mathematical Definition

```
REPELLER: 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

# Repeller 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**: repeller
**Type**: Dynamical Systems / Repeller
**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

```
repeller (○) + 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 and analyzes repellers: invariant sets that push nearby trajectories away in a dynamical system. It clarifies local and global behavior near repelling equilibria and provides diagnostics for robustness and bifurcation effects. Use it to reason about dynamics where instability shapes long-term structure and transition pathways.

How this skill works

The skill inspects a phase space and flow (continuous or discrete) to identify invariant sets with neighborhood trajectories diverging from them. It computes local linearization, checks eigenvalue sign patterns, and assesses global limit set structure and parameter-dependent bifurcations. It also tracks a triadic bookkeeping value (trit = -1) for compositional reasoning in modular analyses.

When to use it

  • You need to classify an invariant set as repelling vs attracting.
  • Analyzing stability near equilibria, periodic orbits, or invariant manifolds.
  • Studying bifurcations where repelling behavior emerges or disappears.
  • Composing dynamical components in a modular or algebraic framework using trit bookkeeping.
  • Designing systems that require intentional instability or divergence control.

Best practices

  • Linearize the flow and inspect eigenvalues for local repulsion before asserting global repeller behavior.
  • Combine local diagnostics with numerical trajectory sampling to confirm neighborhood divergence.
  • Track parameter dependence to detect bifurcations that change repeller status.
  • Use conservative composition rules (trit bookkeeping) when combining skills to preserve algebraic constraints.
  • Document fallback strategies for non-generic or degenerate cases; validate with multiple initial conditions.

Example use cases

  • Classify an equilibrium as a repeller in a model of population dispersal to predict rapid local decline.
  • Analyze a control system to identify unstable manifolds that guide transient responses away from unsafe states.
  • Study bifurcation diagrams to locate parameter ranges where a formerly stable region becomes repelling.
  • Compose with complementary skills (attractor, stability, bifurcation) to map phase space structure in modular analyses.
  • Use trit-aware composition to ensure algebraic conservation when integrating this analysis into larger pipelines.

FAQ

What distinguishes a repeller from an attractor?

A repeller is an invariant set whose nearby trajectories move away over time; an attractor draws nearby trajectories toward it. Local eigenvalue signs typically reflect this difference.

How do I verify repeller behavior numerically?

Linearize at the invariant set, inspect eigenvalues for unstable directions, then simulate nearby initial conditions to confirm that trajectories leave a neighborhood over forward time.