Search

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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]