repeller
Invariant set repelling nearby trajectories
When & Why to Use This Skill
This Claude skill provides advanced capabilities for analyzing repellers within dynamical systems theory. It enables the identification and modeling of invariant sets that repel nearby trajectories, facilitating the study of qualitative behaviors in differential equations, phase space analysis, and complex flows on manifolds.
Use Cases
- Analyzing Unstable Equilibria: Identify and study points or sets in a phase space where trajectories diverge, essential for understanding system boundaries and instability.
- Bifurcation Analysis: Model parameter-dependent qualitative changes in systems, specifically focusing on how repelling sets emerge or transform.
- Computational Mathematics: Integrate with Julia's AlgebraicDynamics.jl to implement compositional dynamical systems and resource-sharing machines.
- Global Structure Mapping: Investigate long-term dynamics and limit sets to understand the high-level architecture of complex mathematical or physical models.
- Theoretical Physics Research: Apply GF(3) triadic composition and Möbius squarefree conditions to analyze non-backtracking geodesic traversals in abstract state spaces.
| 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
- Local behavior: Analysis near equilibria and invariant sets
- Global structure: Long-term dynamics and limit sets
- Bifurcations: Parameter-dependent qualitative changes
- 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
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:
- Prime Path: No state revisited in skill invocation chain
- Möbius Filter: Composite paths (backtracking) cancel via μ-inversion
- GF(3) Conservation: Trit sum ≡ 0 (mod 3) across skill triplets
- 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)