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trajectory

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Trajectory

轨迹

Trit: -1 (MINUS) Domain: Dynamical Systems Theory Principle: Path traced by solution through phase space
Trit: -1 (MINUS) Domain: Dynamical Systems Theory 原理: 解在相空间中经过的轨迹

Overview

概述

Trajectory is a fundamental concept in dynamical systems theory, providing tools for understanding the qualitative behavior of differential equations and flows on manifolds.
轨迹是动力系统理论中的核心概念，为理解微分方程的定性行为和流形上的流提供了工具。

Mathematical Definition

数学定义

TRAJECTORY: Phase space × Time → Phase space
TRAJECTORY: Phase space × Time → Phase space

Key Properties

关键性质

  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
  1. 局部行为: 平衡点与不变集附近的分析
  2. 全局结构: 长期动力学行为与极限集
  3. 分岔: 依赖参数的定性变化
  4. 稳定性: 扰动下的鲁棒性

Integration with GF(3)

与GF(3)的集成

This skill participates in triadic composition:
  • Trit -1 (MINUS): Sinks/absorbers
  • Conservation: Σ trits ≡ 0 (mod 3) across skill triplets
该技能参与三元组合：
  • Trit -1 (MINUS): Sinks/absorbers
  • 守恒性: Σ trits ≡ 0 (mod 3) 跨技能三元组

AlgebraicDynamics.jl Connection

与AlgebraicDynamics.jl的关联

julia
using AlgebraicDynamics
julia
using AlgebraicDynamics

Trajectory as compositional dynamical system

Trajectory as compositional dynamical system

Implements oapply for resource-sharing machines

Implements oapply for resource-sharing machines

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Related Skills

相关技能

  • equilibrium (trit 0)
  • stability (trit +1)
  • bifurcation (trit +1)
  • attractor (trit +1)
  • lyapunov-function (trit -1)
Skill Name: trajectory Type: Dynamical Systems / Trajectory Trit: -1 (MINUS) GF(3): Conserved in triplet composition
  • equilibrium (trit 0)
  • stability (trit +1)
  • bifurcation (trit +1)
  • attractor (trit +1)
  • lyapunov-function (trit -1)
Skill Name: trajectory Type: Dynamical Systems / Trajectory Trit: -1 (MINUS) GF(3): 三元组合中守恒

Non-Backtracking Geodesic Qualification

非回溯测地线资质

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)
条件: μ(n) ≠ 0 (Möbius squarefree)
该技能具备非回溯测地线遍历资质：
  1. 素路径: 技能调用链中无状态重复
  2. 莫比乌斯过滤器: 复合路径（回溯）通过μ反演抵消
  3. GF(3)守恒: 跨技能三元组的Trit和 ≡ 0 (mod 3)
  4. 谱间隙: k-正则扩展满足拉马努金界 λ₂ ≤ 2√(k-1)
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)