Narrow Results

For the class ${𝒦}$ of meromorphic functions outside a compact countable set of essential singularities, we study the dynamics of some functions in ${𝒦}$ for which the unit disc and its complement are invariant. These functions are products and compositions of the function $E_1=exp(\frac{z-1}{z+1})$ with Blaschke products, that include the family $E_n=exp\frac{z^n-1}{z^n+1}$, 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.

We prove that there is a sequence of purely imaginary parameter values ${\lambda }_n$ converging to $0$ such that the Julia set for $z\mapsto {\lambda }_n(z+1/z)$ 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.

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.

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.

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.

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.

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.

There are two natural definitions of the Julia set for complex Hénon maps: the sets $J$ and $J^{\star }$. 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 $J^{\star }$, 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 $J=J^{\star }$ 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 ${ℂ}^2$. 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 $J^{\star }$. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set $J$ were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

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.

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.

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.

Our main theorem establishes that the Fatou set of the functions f_λ,μ(z) = λe^z + 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 f_λ,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.

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.

In this paper, we consider the family of rational maps

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.

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.

The Julia set of the quadratic map f_μ(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.

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 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.