Narrow Results

The continuity of the map φ : c → dim_H(J_c) where J_c is the Julia set of the map f_c(z) = z² + c is discussed in this paper. Using the Hausdorff dimension, a partial order is defined on the Mandelbrot set M. Some properties of irrational points defined by this order are also studied.

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice j = 0.

In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ²ⁿ of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w² + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

Inspired by the study of 3D fractals based on quadratic polynomial maps,¹ this paper presents a novel approach for generating 3D Julia sets by utilizing a family of switching polynomial maps in order to further enrich the form of 3D fractals. Rotational symmetries in the structures of resulting Julia sets with different switching parameters are theoretically analyzed and proved. Experimental results obtained from various parameters and coefficients in the switching maps are provided and discussed in detail. It is hoped that the investigations conducted in this paper can bring on new perspectives for the generalization of 3D fractals.

We extend Kim's complex exponential function, come up with theory about the Julia set of Newton's transformation for general exponential equation, analyze the behavior of the roots of some complex exponential equation, and prove the symmetry, boundedness and embedding topology distribution structure of basins of attraction of the Julia set in theory.

This paper researches the dynamic behavior of a general form of the Fibonacci function, which is a quasi-sine Fibonacci function. It analyses the fixed points of the quasi-sine Fibonacci function on the real axis and the complex plane, and then constructs the Julia set of it using the escape-time method, discovering that the Julia set is fractal and it is on the x-axis symmetry. Using the conception of critical point, the quasi-sine Fibonacci function is generalized. Later the paper examines the dynamic behavior of the generalized quasi-sine Fibonacci function on critical points, and finds that the Mandelbrot set is also on the x-axis symmetry. Finally, it is discovered that there is a jumping phenomenon on the critical points.

In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zⁿ + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zⁿ + c (n ∈ Z, n ≥ 2).

This paper focuses on applying fractal Julia sets to observe the topological properties related to the signs of the real and reactive electric powers. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Julia sets. The study considers three cases: the first study considers the change of real power when the reactive power is fixed; the second study deals with the change of the reactive power when the real power is fixed; and finally, the third study contemplates that both real and reactive powers change. Furthermore, the fractal diagrams of the power in the four quadrants of the complex plane are studied to identify the topological properties for each sign. A qualitative analysis of the diagrams helps identify that complex power loads present some fractal graphic patterns with respect to the signs considered in the different quadrants of the complex planes. The diagrams represented in the complex planes save a relation in the forms and structure with other points studied, concluding that the power is related to other figures in other quadrants. Thus, this result allows a new study of the behavior of power in an electrical circuit by showing a clear relation of the different fractal diagrams obtained by the Julia sets.

This paper presents a preliminary study about a kind of chain coupling system, which we hope could have some enlightened effect for the research on the spatial fractal set of more strongly coupled systems. By analyzing the trajectories of system variables and applying the magnifying or reducing method, the upper bounds of the original and controlled Julia sets from the proposed system are given. Numerical examples are also included to verify the conclusions.

Julia set is one of the most important branches in fractal theory, and has achieved much progress on both theoretical analysis and application research. Nevertheless, most of the existing investigations relied on the systems with known parameters. Few works have been done on the inverse problem about how to determine the system parameter based on a given Julia set’s shape. This work aims to address this issue by starting with the most classical polynomial map. First, by constructing a proper fitness function measuring the Julia sets’ area error, the Julia sets’ parameter estimation is formulated as a kind of optimization problem. According to the known conditions of Julia set to be estimated, the optimization problem is classified into two cases. For one case, when both the shape and size of Julia set are given, we only need to estimate the system’s own parameter $c$. For the other case, when the Julia set has only a shape, we redesign the problem into three dimensions by adding a new scaling factor parameter $\gamma$ and applying the partial coverage principle. Second, a particle swarm optimization (PSO) approach including dynamically adjustment strategy is adopted to solve the proposed optimization problem. At last, we present seven groups of simulations in which both randomly generated Julia sets and those in existing literature (or on the internet) are included. The simulation results verify the effectiveness of the proposed parameter estimation scheme.

This paper proposes a criterion for fractionalizing traditional fractal sets, and reports several fractional generalizations of the Julia set and Mandelbrot set of the classic quadratic map. The discrepancies among those extensions are investigated preliminarily. The influence of the map’s memory effect on its fractal sets is discussed through visualization techniques and dimension analysis.