Lacunarity exponents

Abstract

Many physical processes result in very uneven, apparently random, distributions of matter, characterized by fluctuations of the local density over orders of magnitude. The density of matter in the sparsest regions can have a power-law distribution, with an exponent that we term the lacunarity exponent. We discuss a mechanism which explains the wide occurrence of these power laws, and give analytical expressions for the exponent in some simple models.

Lacunarity transition in a chaotic dynamical system

“…Data from both physical and numerical experiments on non-autonomous chaotic systems indicate that there can be very sparsely occupied regions of phase space. These have previously been investigated for the case of systems which have folds or caustics [11], but in the case of systems with invertible dynamics there is very little previous work.…”

“…The extent to which our results are consistent with this model is considered in our concluding remarks, section 6. A recent paper [11] considered sets arising as attractors of chaotic dynamical systems, and showed evidence that distribution of low densities has a power-law probability density function (PDF). The exponent of this power-law was termed the lacunarity exponent.…”

“…However, the model considered in that work was fundamentally different because its dynamics was many-to-one (as a result of folds or caustics), whereas here we consider a dynamical system which is invertible. The theoretical arguments supporting the power-law described in [11] are critically dependent upon the non-invertible nature of the systems which were considered there. This paper will introduce and analyse a simple model for invertible, non-autonomous, chaotic dynamical systems, such as the surface flow of a chaotically stirred fluid.…”

Meteorological drought lacunarity around the world and its classification

“…Another related measure involves the Lyapunov exponent, which indicates the rate of separation of infinitesimally close trajectories, or involves the inverse, sometimes referred to as Lyapunov time, since it indicates the time expected to become a chaotic trajectory (Boeing, 2016;Kuznetsov, 2016;Gaspard, 2005;Bezruchko and Smirnov, 2010). The Hurst exponent is also related to the fractal dimension of chaotic time series, providing possible long-term memory throughout autocorrelation (Mandelbrot, 1985;Feder, 1988;Yu et al, 2015).…”

“…In this sense, dry pauses of the rainfall can be compared with the gap distribution of fractal objects, which is also known as lacunarity (Martínez et al, 2007;Lucena et al, 2018). The lacunarity analysis is used to characterise "spatial" patterns (such as invariance, density, and heterogeneity) of fractal objects, which represent attractor solutions of nonlinear dynamical systems (Plotnick et al, 1996;Wilkinson et al, 2019). If a time series of precipitation is a solution of the climatic system at a given point, the dryness distribution informs (climate) average features of the system (e.g.…”