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periodic-orbit skill

/skills/periodic-orbit

This skill analyzes periodic orbits in dynamical systems to illuminate long-term behavior and stability across parameter changes.

npx playbooks add skill plurigrid/asi --skill periodic-orbit

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: periodic-orbit
description: Closed trajectory in phase space
version: 1.0.0
---


# Periodic Orbit

**Trit**: 0 (ERGODIC)
**Domain**: Dynamical Systems Theory
**Principle**: Closed trajectory in phase space

## Overview

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

## Mathematical Definition

```
PERIODIC_ORBIT: 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

# Periodic Orbit 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**: periodic-orbit
**Type**: Dynamical Systems / Periodic Orbit
**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

```
periodic-orbit (○) + 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 periodic orbits—closed trajectories in phase space that recur after a fixed period. It helps identify, classify, and reason about local and global behavior of dynamical systems, including stability and bifurcation structure. The focus is on practical diagnostics and compositional use within larger dynamical workflows.

How this skill works

The skill inspects phase space trajectories and returns closed-loop candidates by detecting recurrence, Poincaré section crossings, or periodic solutions of flow maps. It computes stability indicators (Floquet multipliers, Lyapunov exponents) and supports bifurcation checks as parameters vary. It is designed to compose with complementary skills for equilibrium, stability, and attractor analysis.

When to use it

  • When you need to find or verify closed trajectories in ODE or discrete-time systems.
  • To assess the stability and robustness of recurring behavior under perturbation.
  • When exploring bifurcations that create or destroy periodic orbits.
  • As a component of model reduction or skeletonization of long-term dynamics.
  • When composing with other analyses (equilibria, Lyapunov functions, attractors).

Best practices

  • Preprocess trajectories to remove transient behavior before searching for periodicity.
  • Use multiple detection methods (recurrence maps, Poincaré sections, spectral shooting) to cross-validate candidates.
  • Compute Floquet multipliers or linearized return-map eigenvalues to determine orbital stability.
  • Sweep parameters incrementally to reveal bifurcations rather than relying on single-point checks.
  • Combine with redundancy strategies: fallback detection heuristics and alternative section choices to avoid false negatives.

Example use cases

  • Detecting limit cycles in a biochemical oscillator model and classifying them as stable or unstable.
  • Validating periodic solutions returned by numerical continuation and tracking Floquet multipliers across parameter changes.
  • Constructing a reduced dynamical skeleton of a high-dimensional flow by replacing complicated transients with representative periodic orbits.
  • Filtering long simulation data to extract recurring motifs for subsequent machine-learning surrogates.
  • Composing with stability and bifurcation skills to produce robust control strategies that exploit or avoid periodic regimes.

FAQ

How does the skill distinguish transient recurrences from true periodic orbits?

It applies transient truncation, recurrence tolerance thresholds, and consistency checks on multiple cycles (period refinement and return-map convergence) to confirm genuine periodicity.

What stability metrics does it provide?

It reports Floquet multipliers from the linearized return map and can estimate Lyapunov exponents along the orbit to indicate orbital stability and sensitivity.