A novel concept of fractal dimension in deterministic and stochastic Lorenz-63 systems

TL;DR

This paper introduces a new scale-dependent fractal dimension concept to analyze the Lorenz-63 system under noise, revealing how attractor properties vary across scales and distinguishing noise types.

Contribution

It combines adaptive decomposition with extreme value theory to quantify scale-dependent dimensions, advancing the analysis of multi-scale chaotic systems.

Findings

Scale-dependent dimensions vary with focus scale.

The method discriminates between additive and multiplicative noise.

Properties of the invariant set depend on the scale analyzed.

Abstract

Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of 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 as 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 exhibit very similar stochastic attractors 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.