可整合的Toda链中的动态混乱是由时间离散引起的
Carlo Danieli1, Emil A Yuzbashyan2, Boris L Altshuler3
1Physics Department, Sapienza University of Rome, Piazzale Aldo Moro 5, Rome 00185, Italy.
Chaos (Woodbury, N.Y.)
|March 4, 2024
概括
可整合的哈密尔顿动力学的数值模拟,如Toda链,可以因时间分离而引起混乱. 有限时间步骤破坏了整合性,导致有限的利亚普诺夫时间,而较小的步骤接近真正的整合性.
科学领域:
- 计算物理 计算物理
- 动态系统理论 动态系统理论
- 数字分析 数字分析
背景情况:
- 可整合的哈密尔顿系统具有保存量和可预测的长期行为.
- 数字模拟对于研究复杂的动态系统至关重要,但可以引入文物.
- 托达链是可整合的哈密尔顿动力学的经典模型.
研究的目的:
- 研究数值模拟中的时间离散如何影响哈密尔顿系统的整合性.
- 用数值方法量化引发动态混乱的开始.
- 在多达链的长期模拟中分析分解机制.
主要方法:
- 使用Toda链模型进行模拟.
- 雇佣了各种简单的集成器,时间步骤为 τ.
- 测量了利亚普诺夫时间 (TΛ) 作为最大的利亚普诺夫指数 (Λ) 的逆值.
- 结果与不可整合的费米-帕斯塔-乌拉姆-辛链进行了比较.
主要成果:
- 用有限的 τ 进行时间离散破坏了整合性,并在 Toda 链中引发了混乱.
- 利亚普诺夫时间TΛ对于有限的t是有限的,并且随着t接近0.0而分离.
- 模拟表现出分解 (NaN) 的时间 TB ≫ TΛ 由于大位置/动量.
- 鉴定了来自simplectic集成器的周期性驱动作为模拟故障的原因.
结论:
- 可整合的哈密尔顿动态的数值集成可以导致混乱的出现.
- 在数值集成器中选择时间步骤对保存可集成性的影响至关重要.
- 综合集成器虽然具有优势,但可以引入导致模拟不稳定性和故障的工件.
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