Caputo分数阶标准映射中的可调次扩散
Tunable subdiffusion in the Caputo fractional standard map
AI总结：
本文通过数值标度研究证明Caputo分数阶标准映射在强混沌条件下表现出可调次扩散，扩散指数μ(α)随分数阶α从0变化到1，并用现象学解析估计描述结果。
摘要：
Caputo分数阶标准映射（C-fSM）是一个在作用-角变量$(I,θ)$中给出的具有记忆的二维非线性映射。它由参数$K$和$α∈(1,2]$控制，分别调节非线性的强度和Caputo导数的分数阶。在本工作中，我们对强混沌轨道（即当$K\ngg1$时）的平均平方作用$\nleft< I^2 \right>$进行了标度研究。我们数值证明了对于足够大的离散时间$n$，$\nleft< I^2 \right>\npropto n^μ$，其中$0≤μ(α)≤1$。也就是说，我们证明了C-fSM在$1<α<2$时表现出次扩散。具体而言，我们展示了当$α→1$时扩散被抑制，因为$μ(1)=0$，而当$α=2$时恢复标准扩散，此时$μ(2)=1$。我们用一种现象学解析估计来描述我们的数值结果。我们还将C-fSM与Riemann-Liouville fSM和Chirikov标准映射进行了对比。
英文摘要：
The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables $(I,θ)$. It is parameterized by $K$ and $α\in(1,2]$ which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action $\left< I^2 \right>$ along strongly chaotic orbits, i.e. when $K\gg1$. We numerically prove that $\left< I^2 \right>\propto n^μ$ with $0\leμ(α)\le1$, for large enough discrete times $n$. That is, we demonstrate that the C-fSM displays subdiffusion for $1<α<2$. Specifically, we show that diffusion is suppressed for $α\to1$ since $μ(1)=0$, while standard diffusion is recovered for $α=2$ where $μ(2)=1$. We describe our numerical results with a phenomenological analytical estimation. We also contrast the C-fSM with the Riemann-Liouville fSM and Chirikov's standard map.
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