stability
Qualitative behavior of solutions near equilibria
When & Why to Use This Skill
This Claude skill provides advanced capabilities for analyzing the qualitative behavior of dynamical systems near equilibria. It leverages principles from Dynamical Systems Theory to evaluate local stability, global structures, and bifurcations, facilitating deep insights into complex mathematical models and differential equations.
Use Cases
- Stability Analysis: Determining whether a system returns to equilibrium after small perturbations in physics, control theory, or engineering models.
- Bifurcation Study: Identifying critical parameter values where a system's qualitative behavior undergoes a fundamental change, essential for climate or economic forecasting.
- Long-term Dynamics: Predicting the asymptotic behavior and limit sets of complex flows on manifolds for advanced academic research and theoretical modeling.
- Compositional Modeling: Utilizing AlgebraicDynamics.jl integration to analyze resource-sharing machines and the interaction between interconnected dynamical systems.
| name | stability |
|---|---|
| description | Qualitative behavior of solutions near equilibria |
| version | 1.0.0 |
Stability
Trit: 1 (PLUS) Domain: Dynamical Systems Theory Principle: Qualitative behavior of solutions near equilibria
Overview
Stability is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.
Mathematical Definition
STABILITY: 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 (PLUS): Sources/generators
- Conservation: Σ trits ≡ 0 (mod 3) across skill triplets
AlgebraicDynamics.jl Connection
using AlgebraicDynamics
# Stability 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: stability Type: Dynamical Systems / Stability 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:
- 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)