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hopf skill

/skills/hopf

This skill analyzes Hopf bifurcation in dynamical systems to identify limit cycles emerging from equilibria and informs stability and long-term behavior.

npx playbooks add skill plurigrid/asi --skill hopf

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: hopf
description: Bifurcation creating limit cycle from equilibrium
trit: -1
geodesic: true
moebius: "μ(n) ≠ 0"
---

# Hopf

**Trit**: -1 (MINUS)
**Domain**: Dynamical Systems Theory
**Principle**: Bifurcation creating limit cycle from equilibrium

## Overview

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

## Mathematical Definition

```
HOPF: 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

# Hopf 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**: hopf
**Type**: Dynamical Systems / Hopf
**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 Hopf bifurcation — the mechanism where an equilibrium loses stability and a limit cycle emerges. It captures local and global dynamical behavior around equilibria and provides a compact, topological viewpoint useful for analysis and composition. The skill is framed for use in compositional dynamical systems and integrates with algebraic tooling for resource-sharing models.

How this skill works

The skill inspects vector fields near an equilibrium and identifies parameter conditions that produce a pair of complex conjugate eigenvalues crossing the imaginary axis. It classifies resulting behaviors: creation of a stable or unstable limit cycle, changes in attractor structure, and implications for long-term dynamics. It also records trit information (−1) for composition rules used in GF(3)-based skill assemblies.

When to use it

  • Analyzing local bifurcations in ODE models when a pair of eigenvalues becomes marginal
  • Predicting the onset of sustained oscillations in physical, chemical, or biological systems
  • Composing dynamical modules where limit-cycle generation interacts with other attractors
  • Testing robustness of equilibria under parameter variation
  • Integrating with algebraic dynamics frameworks or resource-sharing models

Best practices

  • Linearize around equilibria and compute eigenvalues before applying Hopf criteria
  • Check nondegeneracy and transversality conditions to confirm a true Hopf bifurcation
  • Complement local analysis with numerical continuation to follow emerging limit cycles
  • Record and use the trit (−1) label consistently when composing skills in GF(3) frameworks
  • Verify global effects with phase portraits and Poincaré maps to avoid missing secondary bifurcations

Example use cases

  • Determining when a chemical reaction model will switch from steady state to oscillatory chemistry
  • Designing synthetic biological circuits that require reliable oscillations
  • Predicting rhythm onset in population or epidemiological models as a parameter crosses a threshold
  • Composing a larger dynamical pipeline where Hopf modules balance sink-type components via trit conservation
  • Using AlgebraicDynamics-style tooling to simulate resource-sharing systems that exhibit emergent cycles

FAQ

What distinguishes a Hopf bifurcation from other bifurcations?

A Hopf bifurcation specifically involves a complex conjugate pair of eigenvalues crossing the imaginary axis, leading to a small-amplitude limit cycle emerging from an equilibrium.

When is a detected eigenvalue crossing not a Hopf bifurcation?

If nondegeneracy or transversality conditions fail, or if eigenvalues are real instead of complex conjugates, the event is a different bifurcation and requires alternate analysis.