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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]

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]

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 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]