Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems
Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems
复制标题
DOI：
10.1063/5.0106053
复制
发表时间：
2023-02-01
期刊：
影响因子：
2.9
通讯作者：
Daviaud, F.
中科院分区：
文献类型：
--
作者：
Alberti, T.;Faranda, D.;Daviaud, F.
许多自然系统在不同的尺度上表现出涌现现象，导致了在大尺度上具有确定性混沌特征的标度制度和在小尺度上明显的随机行为。这些特征通常是通过研究基本吸引子的性质来定量研究的，吸引子是紧凑的对象，渐近地承载着系统的轨迹及其在相空间中的不变密度。自然系统的这种多尺度性质使得人们几乎不可能对吸引人的集合有一个清晰的了解。事实上，它跨越了广泛的空间尺度，甚至可能由于非平稳强迫而在时间上发生变化。在这里，我们结合自适应分解方法和极值理论来研究瞬时尺度依赖维度的性质，它是最近被引入来描述湍流和天体物理中这种时间和空间尺度依赖吸引子的。为了定量分析这一度量的性质，我们在受加性或乘性噪声扰动的低维确定性Lorenz-63系统上对其进行了测试。我们证明了不变集的性质取决于我们所关注的尺度，并且尺度相关的维度可以区分加性噪声和乘性噪声，尽管这两种情况在大尺度上具有完全相同的平稳不变度量。所提出的形式论一般有助于研究复杂系统中多尺度涨落的作用，使我们能够处理在广泛的物理系统中表征随机涨落的角色的问题。
Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of deterministic chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set. Indeed, it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here, we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise despite the fact that the two cases have exactly the same stationary invariant measure at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems.