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For the class of meromorphic functions outside a compact countable set of essential singularities, we study the dynamics of some functions in for which the unit disc and its complement are invariant. These functions are products and compositions of the function E 1 = exp (z − 1 z + 1) with Blaschke products, that include the family E n = exp z n − 1 z n + 1 , which are the main topic of the article. Slices of the space of parameters of E n are given with a discussion of their main features. We construct a Poincaré extension of E n to the hyperbolic three-dimensional space and to the 3-sphere. Also we study their dynamics. [ABSTRACT FROM AUTHOR]

We prove that there is a sequence of purely imaginary parameter values λ n converging to 0 such that the Julia set for z ↦ λ n (z + 1 / z) is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 γ 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. [ABSTRACT FROM AUTHOR]

It is of great significance to study the three-dimensional financial system model based on the discrete fractional-order theory. In this paper, the Julia set of the three-dimensional discrete fractional-order financial model is identified to show its fractal characteristics. The sizes of the Julia sets need to be changed in some situations, so it is necessary to achieve control of the Julia sets. In combination with the characteristics of the model, two different controllers based on the fixed point are designed, and the control of the three-dimensional Julia sets is realized by adding the controllers into the model in different ways. Finally, the simulation graphs show that the controllers can effectively control the fractal behaviors. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is of crucial significance to study the infectious disease phenomenon by using the SIRS model and thoughts of Julia set. In this paper, Julia set of the discrete version of the SIRS model is established to analyze the fractal dynamics of this model. Then, controller is designed to change the Julia set. Furthermore, the box-counting dimensions of the controlled Julia sets by selecting different appropriate parameters are computed to show the complexity of the model. Finally, a nonlinear coupling method is introduced to synchronize the Julia sets with different parameters of the same system. Simulation results show the efficacy of these methods. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

There are two natural definitions of the Julia set for complex Hénon maps: the sets and . Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set , under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457-464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457-464], deviating at two points, where substantial dissipativity is used. We show that also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of . VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221-271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on . Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)]. [ABSTRACT FROM AUTHOR]

Lotka-Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods. [ABSTRACT FROM AUTHOR]

The forced Brusselator model is investigated from the fractal viewpoint. A Julia set of the discrete version of the Brusselator model is introduced and control of the Julia set is presented by using feedback control. In order to discuss the relations of two different Julia sets, a coupled item is designed to realize the synchronization of two Julia sets with different parameters, which provides a method to discuss the relation and the changing of two different Julia sets, one Julia set can be changed to be the other. Numerical simulations are used to verify the effectiveness of these methods. [ABSTRACT FROM AUTHOR]

We study a nonanalytic perturbation of the complex quadratic family z ↦ z2 + c in the form of a two-dimensional noninvertible map that has been introduced by Bamón et al. [2006]. The map acts on the plane by opening up the critical point to a disk and wrapping the plane twice around it; points inside the disk have no preimages. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. For parameters away from the complex quadratic family we define a generalized notion of the Julia set as the basin boundary of infinity. We are interested in how the Julia set changes when saddle points along with their stable and unstable sets appear as the perturbation is switched on. Advanced numerical techniques enable us to study the interactions of the Julia set with the critical set and the (un)stable sets of saddle points. We find the appearance and disappearance of chaotic attractors and dramatic changes in the topology of the Julia set; these bifurcations lead to three complicated types of Julia sets that are given by the closure of stable sets of saddle points of the map, namely, a Cantor bouquet and what we call a Cantor tangle and a Cantor cheese. We are able to illustrate how bifurcations of the nonanalytic map connect to those of the complex quadratic family by computing two-parameter bifurcation diagrams that reveal a self-similar bifurcation structure near the period-doubling route to chaos in the complex quadratic family. [ABSTRACT FROM AUTHOR]

Our main theorem establishes that the Fatou set of the functions fλ,μ(z) = λez + μ/z contains a two-cycle of Baker domains {F∞, F0} if Re(λ) < 0 and |Im(λ)| < 1/2|Re(λ)|. We show that under every point in F∞ tends to infinity and in F0 tends to zero. Moreover, if |Im(λ)| < 1/2|Re(λ)| - 4, the set F∞ contains infinitely many critical points of fλ,μ and F0 contains infinitely many critical values; also, infinitely many critical values of are contained in both F∞ and F0. Finally, the images of the Baker domains are displayed for some parameters. [ABSTRACT FROM AUTHOR]

The dynamical and fractal behaviors of the complex perturbed rational maps are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider the family of rational maps where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids. [ABSTRACT FROM AUTHOR]

Escape time algorithm is a classical algorithm to calculate the Julia sets, but it has the disadvantage of dull color and cannot record the iterative process of the points. In this paper, we present the equipotential point algorithm to calculate the Julia sets by recording the strike frequency of the points in the iterative process. We calculate and analyze the Julia sets in the complex plane by using this algorithm. Finally, we discuss the iteration trajectory of a single point. [ABSTRACT FROM AUTHOR]

In the present work, we extend the results of the study of the structural stability of the Julia sets of noise-perturbed complex quadratic maps in the presence of dynamic and output noise both for the additive and the multiplicative cases. The critical values of the strength of the noise for which the Julia set of a family of noise-perturbed complex quadratic maps completely loses its original Julia structure were also calculated. Using graphical tools we demonstrate how one can localize the regions of the Julia sets that are affected by the presence of noise in each case. Finally, two numerical invariants for the Julia set of noise-perturbed complex quadratic maps are proposed for the study of the noise effect. [ABSTRACT FROM AUTHOR]

The Julia set of the quadratic map fμ(z) = μz(1 - z) for μ not belonging to the Mandelbrot set is hyperbolic, thus varies continuously. It follows that a continuous curve in the exterior of the Mandelbrot set induces a continuous family of Julia sets. The focus of this article is to show that this family can be obtained explicitly by solving the initial value problem of a system of infinitely coupled differential equations. A key point is that the required initial values can be obtained from the anti-integrable limit μ → ∞. The system of infinitely coupled differential equations reduces to a finitely coupled one if we are only concerned with some invariant finite subset of the Julia set. Therefore, it can be employed to find periodic orbits as well. We conduct numerical approximations to the Julia sets when parameter μ is located at the Misiurewicz points with external angle 1/2, 1/6, or 5/12. We approximate these Julia sets by their invariant finite subsets that are integrated along the reciprocal of corresponding external rays of the Mandelbrot set starting from the anti-integrable limit μ = ∞. When μ is at the Misiurewicz point of angle 1/128, a 98-period orbit of prescribed itinerary obtained by this method is presented, without having to find a root of a 298-degree polynomial. The Julia sets (or their subsets) obtained are independent of integral curves, but in order to make sure that the integral curves are contained in the exterior of the Mandelbrot set, we use the external rays of the Mandelbrot set as integral curves. Two ways of obtaining the external rays are discussed, one based on the series expansion (the Jungreis-Ewing-Schober algorithm), the other based on Newton's method (the OTIS algorithm). We establish tables comparing the values of some Misiurewicz points of small denominators obtained by these two algorithms with the theoretical values. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z′ = z2 - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x2 - c with x a real variable. Here, the second part deals with an embedding of TZ into the two-dimensional noninvertible map $\overline{T}: x^{\prime }=x^{2}+y-c$; y′ = γ y + 4x2y, γ ≥ 0. For $\gamma = 0, \overline{T}$ is semiconjugate to TZ in the invariant half plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets, from a study of the basin boundary of the attractor located on y = 0. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family $z_{n+1} = z_n^2 + c_i, i = 1,2$ and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two. Some examples are presented. [ABSTRACT FROM AUTHOR]

We extend Kim's complex exponential function and come up with a theory about Julia sets of Newton method for general exponential equation. We analyze the behavior of the roots of some complex exponential equation, and prove the Julia Set's symmetry, boundedness and embedding topology distribution structure of attraction regions in theory. [ABSTRACT FROM AUTHOR]

Properties of the different configurations of Julia sets J, generated by the complex map TZ: z′ = z2 - c, are revisited when c is a real parameter, -1/4 < c < 2. This is done from a detailed knowledge of the fractal bifurcation organization "box-within-a-box", related to the real Myrberg's map T: x′ = x2 - λ, first described in 1975. Part I of this paper constitutes a first step, leading to Part II dealing with an embedding of TZ into the two-dimensional noninvertible map $\overline{T}: x^{\prime} =x^{2} + y - c; y^{\prime} = \gamma y + 4x^{2}y, \gamma \geq 0$. For γ = 0, $\overline{T}$ is semiconjugate to TZ in the invariant half-plane (y ≤ 0). With a given value of c, and with γ decreasing, the identification of the global bifurcations sequence when γ → 0, permits to explain a route toward the Julia sets. With respect to other papers published on the basic Julia and Fatou sets, Part I consists in the identification of J singularities (the unstable cycles and their limit sets) with their localization on J. This identification is made from the symbolism associated with the "box-within-a-box" organization, symbolism associated with the unstable cycles of J for a given c-value. In this framework, Part I gives the structural properties of the Julia set of TZ, which are useful to understand some bifurcation sequences in the more general case considered in Part II. Different types of Julia sets are identified. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a rational iteration method which can be used as a root-finding algorithm for almost every polynomial. It has no nonrepelling extraneous fixed point in the complex plane and is generally convergent for both quadratic and cubic polynomials. Then some properties of this algorithm are given. By the aid of computer, we produce pictures of the Julia sets for the iterations of some polynomials. Numerical results show that it is a root-finding method with convergence order the same as Halley's method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the family of rational maps given by \[ F_\lambda(z) = z^n + \frac{\lambda}{z} \] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z2 + c)eQ(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described. [ABSTRACT FROM AUTHOR]

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components. [ABSTRACT FROM AUTHOR]

In this paper we describe several new types of invariant sets that appear in the Julia sets of the complex exponential functions Eλ(z) = λez where λ ∈ ℂ in the special case when λ is a Misiurewicz parameter, so that the Julia set of these maps is the entire complex plane. These invariant sets consist of points that share the same itinerary under iteration of Eλ. Previously, the only known types of such invariant sets were either simple hairs that extend from a definite endpoint to ∞ in the right half plane or else indecomposable continua for which a single hair accumulates everywhere upon itself. One new type of invariant set that we construct in this paper is an indecomposable continuum in which a pair of hairs accumulate upon each other, rather than a single hair having this property. The second type consists of an indecomposable continuum together with a completely separate hair that accumulates on this continuum. [ABSTRACT FROM AUTHOR]

The continuity of the map φ : c → dimH(Jc) where Jc is the Julia set of the map fc(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. [ABSTRACT FROM AUTHOR]

For a nonconstant function F and a real number h∈]0, 1] the relaxed Newton's method NF,h of F is an iterative algorithm for finding the zeroes of F. We show that when relaxed Newton's method is applied to complex function F(z)=P(z)eQ(z), where P and Q are polynomials, the basin of attraction of a root of F has finite area if the degree of Q exceeds or equals 3. The key point is that NF,h is a rational map with a parabolic fixed point at infinity. [ABSTRACT FROM AUTHOR]

Let Q denote a quadratic polynomial and A∞ the super-attracting basin of Q at the point ∞ on the Riemann sphere $\hat{\mathbb C}$. There exists a unique Riemann mapping Φ from the open disk $D=\{z\in \hat{\mathbb C}: |z|>1\}$ onto A∞ such that Φ(∞)=∞, Φ'(∞)=1 and Φ-1 conjugates Q:A∞→A∞ to the squaring map S:D→D:z↦z2. In this paper, we show if Q is real and infinitely renormalizable of bounded type then the continuous extension of Φ to the closed disk cannot have any Hölder continuity on the boundary circle ${\mathbb S}^1$. [ABSTRACT FROM AUTHOR]

We consider the family of entire transcendental maps given by F[sub λ,m](z)=λz[sup m]exp(z) where m≥2. All functions F[sub λ,m] have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle. [ABSTRACT FROM AUTHOR]

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map. [ABSTRACT FROM AUTHOR]