INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100, 00133 Roma, Italy.
Laboratoire des Sciences du Climat et de l'Environnement, CEA Saclay l'Orme des Merisiers, UMR 8212 CEA-CNRS-UVSQ, Université Paris-Saclay, and IPSL, 91191 Gif-sur-Yvette, France.
Chaos. 2023 Feb;33(2):023144. doi: 10.1063/5.0106053.
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.
许多自然系统在不同尺度上表现出涌现现象，导致在大尺度上具有确定性混沌特征的标度律和在小尺度上具有明显随机行为的标度律。这些特征通常通过研究基础吸引子的特性来定量研究，吸引子是一个紧凑的物体，在相空间中具有系统轨迹的不变密度，并渐近地容纳它们。自然系统的这种多尺度性质使得实际上不可能清楚地了解吸引集。事实上，它跨越了广泛的空间尺度，甚至由于非平稳强迫而随时间变化。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相关维数的特性，该维数最近被引入到湍流和天体物理学中以描述这种时间和空间尺度相关的吸引子。为了对该度量标准的特性进行定量分析，我们在添加了加性或乘性噪声的著名低维确定性 Lorenz-63 系统上对其进行了测试。我们证明了不变集的特性取决于我们关注的尺度，并且尽管两种情况在大尺度上具有完全相同的平稳不变测度，但尺度相关维度可以区分加性噪声和乘性噪声。所提出的形式可以一般有助于研究复杂系统中多尺度波动的作用，使我们能够处理在广泛的物理系统中表征随机波动作用的问题。
