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We study the group TA of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). We prove that TA is generated by a copy of Thompson's group F and a copy of Thompson's group T, hence it is finitely generated. Then we study the commutator subgroup [TA; TA], proving that the abelianization of TA is isomorphic to Z and that [TA; TA] is simple, finitely generated and acts 2-transitively on the so-called components of the airplane limit space. Moreover, we show that TA is contained in T and contains a natural copy of the basilica rearrangement group TB studied by Belk and Forrest (2015). [ABSTRACT FROM AUTHOR]

A neutrosophic fuzzy set that generalizes the classical set is represented by a closed interval [0,1]. Let us generalize the fuzzy set to the Neutrosophic set, which is defined as three membership functions between interval ]-0,1+ [. This paper examines the bird collision problem within a more grounded Neutrosophic framework. The collision between the bird and the aircraftis represented in the Neutrosophic domain as a stationary point because it occurs at a distance too great for any other auditory signal to discern. Numerous methods exist for solving the airplane problem, including a Julia set in complex numbers. Bird collisions resulting from aircraft crashes add complication. An aviation signalis utilized to avoid anaccident in a bird strike or bird collision.

A neutrosophic fuzzy set that generalizes the classical set is represented by a closed interval [0,1]. Let us generalize the fuzzy set to the Neutrosophic set, which is defined as three membership functions between interval]-0,1+ [. This paper examines the bird collision problem within a more grounded Neutrosophic framework. The collision between the bird and the aircraft is represented in the Neutrosophic domain as a stationary point because it occurs at a distance too great for any other auditory signal to discern. Numerous methods exist for solving the airplane problem, including a Julia set in complex numbers. Bird collisions resulting from aircraft crashes add complication. An aviation signal is utilized to avoid an accident in a bird strike or bird collision. [ABSTRACT FROM AUTHOR]

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families. [ABSTRACT FROM AUTHOR]

When aiming to achieve consistency in fractal characteristics between different models, it is crucial to consider the synchronization of Julia sets. This paper studies the synchronization of Julia sets in three-dimensional discrete financial models. First, three-dimensional discrete financial models with different model parameters are proposed and their Julia sets are presented. According to the model forms, two kinds of synchronous couplers that can achieve synchronization of Julia sets between different models are designed by changing the synchronization parameters. The proposed synchronization method is theoretically derived and the efficiency of different synchronous couplers are compared. Finally, the effectiveness is verified by Julia sets graphics. This method has reference value for theoretical research into financial models in the field of fractals. [ABSTRACT FROM AUTHOR]

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures. [ABSTRACT FROM AUTHOR]

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]

The majority of fractals' dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of "escape" is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine T c (z) = a s i n (z r) + b z + c and complex exponential T c (z) = a e z r + b z + c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters 'a' and 'b', and the parameters involved in the series expansion of the sine and exponential functions. [ABSTRACT FROM AUTHOR]

In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between T(r, f) and log M(r, f) when the modulus of variable |z| = r is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results. [ABSTRACT FROM AUTHOR]

We show the structure of the parameter space for a family of rational maps containing Blaschke products. Through numerical simulations using the orbit of a single critical point, we reveal the existence of infinitely many Mandelbrot-like sets along the unit circle, as well as eight-like structures in other regions of parameter space. We pose some open questions related to the parameter space of these functions.

To investigate the degree d connectedness locus, Thurston [ On the geometry and dynamics of iterated rational maps , Complex Dynamics, A K Peters, Wellesley, MA, 2009, pp. 3–137] studied \sigma _d-invariant laminations , where \sigma _d is the d-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials f(z) = z^2 +c. In the spirit of Thurston's work, we consider the space of all cubic symmetric polynomials f_\lambda (z)=z^3+\lambda ^2 z in a series of three articles. In the present paper, the first in the series, we construct a lamination C_sCL together with the induced factor space \mathbb {S}/C_sCL of the unit circle \mathbb {S}. As will be verified in the third paper of the series, \mathbb {S}/C_sCL is a monotone model of the cubic symmetric connectedness locus , i.e. the space of all cubic symmetric polynomials with connected Julia sets. [ABSTRACT FROM AUTHOR]

We will give a complete description of the dynamics of the rational map $N_{F_{M_c}}(z)=\frac{3z^4-2z^3+c}{4z^3-3z^2+c}$ where c is a complex parameter. These are rational maps $N_{F_{M_c}}$ arising from Newton's method. The polynomial of Newton iteration function is obtained from singularly perturbed of the Multibrot set polynomial.

In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures.

The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine Tc(z)=asin(zr)+bz+c and complex exponential Tc(z)=aezr+bz+c functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘a’ and ‘b’, and the parameters involved in the series expansion of the sine and exponential functions.

In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine the alteration in generated graphical images and study the consequences of underlying parameters on the variation of dynamics, colour, time of generation, and shape of generated fractals. The black points in the obtained fractals are the "non-chaotic" points and the dynamical behaviour in the black area is easily predictable. The coloured points are the points that "escape", that is, they tend to infinity under one of iterative methods based on a four-step fixed-point iteration scheme extended with s-convexity. The different colours tell us how quickly a point escapes. The order of escaping of coloured points is red, orange, yellow, green, blue, and violet, that is, the red point is the fastest to escape while the violet point is the slowest to escape. Mostly, these generated fractals have symmetry. The Julia set, where we find all of the chaotic behaviour for the dynamical system, marks the boundary between these two categories of behaviour points. The Mandelbrot set, which was originally observed in 1980 by Benoit Mandelbrot and is particularly important in dynamics, is the collection of all feasible Julia sets. It perfectly sums up the Julia sets. [ABSTRACT FROM AUTHOR]

We show that for each d ∈ (0 , 2 ] there exists a meromorphic function f such that the inverse function of f has three singularities and the Julia set of f has Hausdorff dimension d. [ABSTRACT FROM AUTHOR]

We study the dynamics of a meromorphic perturbation of the family λsinz\lambda \sin z by adding a pole at zero and a parameter μ\mu , that is, fλ,μ(z)=λsinz+μ/z{f}_{\lambda ,\mu }\left(z)=\lambda \sin z+\mu \hspace{-0.08em}\text{/}\hspace{-0.08em}z, where λ,μ∈C⧹{0}\lambda ,\mu \in {\mathbb{C}}\hspace{-0.16em}\setminus \hspace{-0.16em}\left\{0\right\}. We study some geometrical properties of fλ,μ{f}_{\lambda ,\mu } and prove that the imaginary axis is invariant under fn{f}^{n} and belongs to the Julia set when ∣λ∣≥1| \lambda | \ge 1. We give a set of parameters (λ,μ)\left(\lambda ,\mu ), such that the Fatou set of fλ,μ{f}_{\lambda ,\mu } has two super-attracting domains. If λ=1\lambda =1 and μ∈(0,2)\mu \in \left(0,2), the Fatou set of f1,μ{f}_{1,\mu } has two attracting domains. Also, we give parameters λ,μ\lambda ,\mu such that ±π/2\pm \pi \hspace{-0.08em}\text{/}\hspace{-0.08em}2 are fixed points of fλ,μ{f}_{\lambda ,\mu } and the Fatou set of fλ,μ{f}_{\lambda ,\mu } contains attracting domains, parabolic domains, and Siegel discs, we present examples of these domains. This paper closes with an example of fλ,μ{f}_{\lambda ,\mu }, where the Fatou set contains two types of domains, for λ,μ\lambda ,\mu given.

The limit sets of zeros of the partition function for $\lambda $ -state Potts models on diamond-like hierarchical lattice are the Julia sets of functions in a family of rational functions. In this paper, the consensus problem of Julia sets generated by $\lambda $ -state Potts models on diamond-like hierarchical lattice is studied. Two types of the consensus problem of Julia sets are considered, one is with a leader and the other is with no leaders. Based on these two types, two different control protocols are proposed respectively to make systems achieve consensus of Julia sets. The simulations confirm the efficacy of control protocols.

Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here $ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $ where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.

In this research, we look at the Julia set patterns that are linked to the entire transcendental function f (z) = a e z n + b z + c , where a , b , c ∈ C and n ≥ 2 , using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function. [ABSTRACT FROM AUTHOR]

Let g (z) = ∫ 0 z p (t) exp (q (t)) d t + c where p, q are polynomials and c ∈ C , and let f be the function from Newton's method for g. We show that under suitable assumptions on the zeros of g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that f n (z) converges to zeros of g almost everywhere in C if this is the case for each zero of g ′ ′ that is not a zero of g or g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero. [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]

We study the dynamics of a generic endomorphism f of an Oka–Stein manifold X. Such manifolds include all connected linear algebraic groups and, more generally, all Stein homogeneous spaces of complex Lie groups. We give several descriptions of the Fatou set and the Julia set of f. In particular, we show that the Julia set is the derived set of the set of attracting periodic points of f and that it is also the closure of the set of repelling periodic points of f. Among other results, we prove that f is chaotic on the Julia set and that every periodic point of f is hyperbolic. We also give an explicit description of the "Conley decomposition" of X induced by f into chain-recurrence classes and basins of attractors. For X = C , we prove that every Fatou component is a disc and that every point in the Fatou set is attracted to an attracting cycle or lies in a dynamically bounded wandering domain (whether such domains exist is an open question). [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]

Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results. [ABSTRACT FROM AUTHOR]

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function hl(z) = z2 +l. Their generalization was based on the composition of hl with the Möbius transformation m(z) = 1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of m(hl(z)) (at z = 0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation. [ABSTRACT FROM AUTHOR]

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function h(z) = z2 + l. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented. [ABSTRACT FROM AUTHOR]

Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions h(z)=zn+c, h(z)=sin(zn)+c and h(z)=ezn+c, n≥2,c∈C. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters.

In this paper, a Julia multiband fractal is designed for use in wireless communication applications. The proposed antenna contains Julia set embedded on FR epoxy substrate with thickness 1mm. Perturbations in the Julia fractal is also studied. The antenna has different characteristics and values because of perturbations. [ABSTRACT FROM AUTHOR]

This work focuses on a kind of fractals Parrondo's paradoxial phenomenon "deiconnected+diconnected=connected" in an alternated superior complex system zn+1 = β b(zn² + ci) + (1-β) zn, i = 1, 2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the Mandelbrot and Julia sets by using Picard-Mann iteration procedure. Escape criteria is established which plays an important role to generate Mandelbrot and Julia sets. Also, numerous graphical pictures of these sets have been visualized and certain examples have been recognized. Presented results shows that fractal images generated by Picard-Mann iteration procedure are entirely different from those generated in Mann orbit.

This research was conducted to gain some patterns of fractals Julia set and Seirpinski that applied on Batik then creating Batik that has many varied motives and multifaceted. There are three steps in formulating the patterns of fractals of Julia set and Seirpinski. First, build the fractals by analyzing the function of fractals Julia set and determine the plane’s coordinate which you want to use. In this case, we use square and rectangle which will be created by using fractals patterns Seirpinski. Second, create a batik motives from fractals pattern Julia set and Seirpinski by using geometry transformation. The geometry transformation which will be used are translation, dilatation, reflection, and rotation. The last, combine some batik motives which were created by using image processing. It was summation of two images processing. The result is batik motives that has many variated and multifaceted.

In this research, we look at the Julia set patterns that are linked to the entire transcendental function f(z)=aezn+bz+c, where a,b,c∈C and n≥2, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function.

This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]

We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property of q-differential equations of higher order in this paper. By giving q-varying variations, we identify the shape of the approximate roots of q-Bernoulli polynomials, a solution of q-differential equations of higher order, and find several conjectures associated with them. Furthermore, based on q-Bernoulli polynomials, we create a Mandelbrot set and a Julia set to find a variety of related figures.

In recent years, researchers have studied the use of different iteration processes from fixed point theory for the generation of complex fractals. Examples are the Mann, the Ishikawa, the Noor, the Jungck-Mann and the Jungck-Ishikawa iterations. In this paper, we present a generalisation of complex fractals, namely Mandelbrot, Julia and multicorn sets, using the Jungck-CR implicit iteration scheme. This type of iteration does not reduce to any of the other iterations previously used in the study of complex fractals; thus, this generalisation gives rise to new fractal forms. We prove a new escape criterion for a polynomial of the following form zm − az + c, where a, c ∈ ℂ, and present some graphical examples of the obtained complex fractals. [ABSTRACT FROM AUTHOR]

We consider a solution to the problem of copyright protection by embedding a secret message into a still color or grayscale image using algebraic fractals. The peculiarity of the proposed steganographic method is the use of a two-dimensional fractal image of algebraic type as an intermediate cover image. For this purpose, it is proposed to use a fractal image of algebraic type in the form of a Julia set as an intermediate cover image. An attacker will not be able to generate an identical fractal image without the exact value of some complex number , which is agreed in advance between the sender and the recipient, as well as a number of other parameters, which makes the proposed method resistant to attacks. The advantage of the proposed algorithm is the ability to extract the watermark without knowledge of the original cover image, since the intermediate container for the watermark is a fractal. In this paper we consider fractal image generation techniques. Then generated fractals are used as cover image in steganography system. Watermark embeds in blue component of JPEG image. To assess the quality of obtained results, a quality assessment was made based on the following metrics: normalized mean square error (NMSE) and peak signal-to-noise ratio (PSNR), expressed in decibels. The results of the original cover image and stego image, as well as the original and extracted watermark, are presented. Analysis of the results of the evaluation of the quality of the original cover image and watermarked image, as well as the original and extracted watermark showed that the proposed algorithm provides high quality of hiding confidential information. The use of fractals as an intermediate cover image for the extraction of watermark allows ensuring the extraction practically without losses, while the stego image visually does not differ from the container containing classified information.

The visual beauty, self-similarity, and complexity of Mandelbrot sets and Julia sets have made an attractive field of research. One can find many generalizations of these sets in the literature. One such generalization is the use of results from fixed-point theory. The aim of this paper is to provide escape criterion and generate fractals (Julia sets and Mandelbrot sets) via CR iteration scheme with s-convexity. Many graphics of Mandelbrot sets and Julia sets of the proposed three-step iterative process with s-convexity are presented. We think that the results of this paper can inspire those who are interested in generating automatically aesthetic patterns.

In today’s world, fractals play an important role in many fields, e.g., image compression or encryption, biology, physics, and so on. One of the earliest studied fractal types was the Mandelbrot and Julia sets. These fractals have been generalized in many different ways. One of such generalizations is the use of various iteration processes from the fixed point theory. In this paper, we study the use of Jungck-CR iteration process, extended further by the use of $s$ -convex combination. The Jungck-CR iteration process with $s$ -convexity is an implicit three-step feedback iteration process. We prove new escape criteria for the generation of Mandelbrot and Julia sets through the proposed iteration process. Moreover, we present some graphical examples obtained by the use of escape time algorithm and the derived criteria.

In today’s world, complex patterns of the dynamical framework have astounding highlights of fractals and become a huge field of research because of their beauty and unpredictability of their structure. The purpose of this paper is to visualize anti-Julia sets, tricorns, and multicorns by means of the Noor iteration with s-convexity. Various patterns are displayed to investigate the geometry of antifractals for antipolynomial $\overline {z}^{k+1}+c$ of complex polynomial $z^{k+1}+c$ , for $k\geq 1$ in Noor orbit with s-convexity.

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods.

The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z \to (1+ \mu) \exp (iz)$, $g : z \to \big(1+ \mu |z - z^*|\big) \exp (iz)$, $h : z \to \big(1+ \mu (z - z^* )\big) \exp (iz)$, with $z,\mu \in \mathbb{C}$, $z^* : \exp (iz^*) = z^*$. For a quasilinear map g having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of "fork" type and "saddle"-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of "valleys" on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of "period doubling" is noted (in one case along the path $39\to 3$, in the other along the path $17\to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.

The Lotka-Volterra model plays an important role in the research area of population biology. This work presents the analysis of dynamical behaviours of a kind of three-species Gause-Lotka-Volterra (GLV for short) system from the viewpoint of fractals. First, the definition of Julia set which describes the initial distribution rule of the three species' densities is introduced. Second, the gradient control method which contains both giant parameter and state feedback is applied to realize the boundary control of the initial fractals area of three coexisting species. Third, we consider the upper bound of the controlled Julia set from a kind of weakly-coupled GLV system, i.e. NPZsystem, by analysing the growth pattern of the initial species. Finally, the nonlinear coupling terms are designed to realize the synchronization of two Julia sets, with the result that the dynamic behaviors of the controlled system can be guided to an ideal one. Numerical examples are included to verify the conclusions of the theoretical investigations. [ABSTRACT FROM AUTHOR]

In this paper, we define (p,q)-cosine and sine sigmoid polynomials. Based on this, the properties of each polynomial, and the structure and assumptions of its roots, can be identified. Properties can also be determined by the changes in p and q.

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number s ∈ (1 , 2) , we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly s. [ABSTRACT FROM AUTHOR]

The objective of this paper is to generate fractals for complex valued polynomials using Jungck-SP orbit. We present an algorithm to generate fractals. With the help of this algorithm, we generate such beautiful graphics which may be useful for graphic designers. In this paper, we extend and improve the corresponding results of Kang et al. (2015) and Kumari et al. (2017) existing in the literature. One can further generalize our results and derive a new escape criterion to generate some more beautiful fractals. [ABSTRACT FROM AUTHOR]