Finite-size local dimension as a tool or extracting geometrical properties of attractors of dynamical systems

Abstract

Local dimension computed using extreme value theory (EVT) is usually used as a tool to infer dynamical properties of a given state ζ of the chaotic attractor of the system. The dimension computed in this way is also known as the pointwise dimension in dynamical systems literature and is defined using a limit for an infinitely small neighborhood in the phase space around ζ. Since it is numerically impossible to achieve such a limit, and because dynamical systems theory predicts that this local dimension is almost constant over the attractor, understanding the properties of this tool for a finite scale R is crucial. We show that the dimension can considerably depend on R, and this view differs from the usual one in geophysics literature, where it is often considered that there is one dimension for a given dynamical state or process. We also systematically assess the reliability of the computed dimension given the number of points to compute it. This interpretation of the R dependence of the local dimension is illustrated on the Lorenz 63 system not only for ρ = 28, but also in the intermittent case where ρ = 166.5. The latter case shows how the dimension can be used to infer some geometrical properties of the attractor in phase space. The Lorenz 96 system with n = 50 dimensions is also used as a higher-dimension example. A dataset of radar images of precipitation (the RADCLIM dataset) is finally considered, with the goal of relating the computed dimension to the (in)stability of a given rain field.