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To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.

This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs.

The cello suites of Johann Sebastian Bach exhibit several types of power-law scaling, the best examples of which can be considered fractal in nature. This article examines scaling with respect to the characteristics of melodic interval and its derivative, melodic moment. A new and effective method for pitch-related analysis is described and then applied to a selection of the 36 pieces that comprise the six cello suites.

Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has that Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.

For a family of networks ${\{G_n\}}_{n\ge 1}$, we define the Hausdorff dimension of ${\{G_n\}}_{n\ge 1}$ inspired by the Frostman’s characteristics of potential for Hausdorff dimension of fractals on Euclidean spaces. We prove that our Hausdorff dimension of the touching networks is $logm/logN.$ Our definition is quite different from the fractal dimension defined for real-world networks.

In 2023, Xi et al. introduced the Hausdorff dimension of a family of networks which inspired by the potential theoretic methods in fractal geometry. In this paper, we will construct a class of colored substitution networks and obtain its Hausdorff dimension using the self-similarity.