On a topological closeness of perturbed Mandelbrot sets

In the present work we propose a numerical and visual tool for the study of the deformation of the Mandelbrot sets of perturbed Mandelbrot maps by noise in comparison with the original Mandelbrot set. Further, by employing these numerical tools, we support the invariance of the Mandelbrot set of a noise-perturbed Mandelbrot map under different noise realizations. Finally, we provide evidence for the non-fractal structure of the Mandelbrot set of a noise-perturbed Mandelbrot map.     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

Is the Mandelbrot set computable?

This work is concerned with the question whether the Mandelbrot set is computable. The computability notions that we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two‐sided distance function of the Mandelbrot set is computable if the famous hyperbolicity conjecture is true. We also formulate the question whether the distance function of the Mandelbrot set is computable in terms of the escape time. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Perfect subspaces of quadratic laminations Dedicated to the Memory of Lei Tan

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady(1984) and, separately, by Thurston(1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be "unrenormalized" to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.     

An epicycloidal boundary of the main component in the degree-n bifurcation set

It is known that the parametric boundary equation for the main component in the Mandelbrot set represents a cardioid. We derive an epicycloidal boundary equation of the main component in the degree-n bifurcation set by extending the parameter which describes the cardioid in the Mandelbrot set. Computational results as well as some useful properties are presented together with the programming source codes written inMathematica. Various boundaries are displayed for 2≤n≤7 and show a good agreement with the theory presented here. The known boundary equation enables us to significantly reduce the construction time for the degree-n bifurcation set.     

Perfect subspaces of quadratic laminations

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady (1984) and, separately, by Thurston (1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by ?pinching" the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be ?renormalized" to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.     

On perturbations of the Mandelbrot map

In this work, we propose new applications of analytic and non-analytic perturbations of the Mandelbrot map as expressed in a two-parameter deformation family of it. The influence of alternative applications of noise for specific choices of a Mandelbrot set is also provided. Hence, we are in a position of examining the stability of this set under stochastic perturbations.     

Connectivity of the Mandelbrot set for the family of renormalization transformations

We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.     

On a topological closeness of perturbed Julia sets

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.     

On 2D generalization of Higuchi’s fractal dimension

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface. This method represents a generalization of Higuchi’s method for calculating fractal dimension of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we construct Weierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method. The 2D fractal analysis method was applied to the set of histological images collected during direct shoot organogenesis from leaf explants. The efficiency of the proposed method in differentiating phases of organogenesis is proved.     

Hyperbolicity,stability and monodromy of dynamical systems

We propose a rigorous computational method for proving uniform hyperbolicity of dynamical systems. Besides finding structurally stable parameters, the algorithm can also be applied for the computation of the monodromy of dynamical systems. With this algorithm, we prove that the topology of the 2-dimensional generalization of the Mandelbrot set is totally different from that of the original Mandelbrot set. Furthermore, we show that the monodromy of the complex Hénon map can be used to determine the dynamics of the real Hénon map. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The area of the Mandelbrot set

Summary We obtain upper bounds for the area of the Mandelbrot set. An effective procedure is given for computing the coefficients of the conformal mapping from the exterior of the unit circle onto the exterior of the Mandelbrot set. The upper bound is obtained by computing finitely many of these coefficient and applying Green's Theorem. The error in such calculations is estimated by deriving explicit formulas for infinitely many of the coefficients and comparing.Partially supported by a grant from the National Science Foundation     

A general view of pseudoharmonics and pseudoantiharmonics to calculate external arguments of Douady and Hubbard

Harmonics give us a compact formula and a powerful tool in order to calculate the external arguments of the last appearance hyperbolic components and Misiurewicz points of the Mandelbrot set in some particular cases. Antiharmonics seem however to have no application. In this paper, we give a general view of pseudoharmonics and pseudoantiharmonics, as a generalization of harmonics and antiharmonics. Pseudoharmonics turn out to be a more powerful tool than harmonics since they allow the calculation of external arguments of the Mandelbrot set in many more cases. Likewise, unlike antiharmonics, pseudoantiharmonics turn out to be a powerful tool to calculate external arguments of the Mandelbrot set in some cases.     

复迭代映射z_(n 1)= c的 广义Mandelbrot集研究

研究了复迭代映射ｚ（ｎ＋１）＝／ｚｎｍ＋ｃ的广义Ｍａｎｄｅｌｂｒｏｔ集,指出其关于实轴是对称的,并且具有ｍ＋１次的旋转对称性,得出周期轨道的稳定性条件及一周期轨道的稳定区域的边界方程．利用逃逸时间算法和周期点查找的算法构造Ｍａｎｄｅｌｂｒｏｔ集,可以更清楚地了解Ｍａｎｄｅｌｂｒｏｔ集的结构．     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Determination of Mandelbrot Sets Hyperbolic Component Centres

In this work we present a very fast and parsimonious method to calculate the centre coordinates of hyperbolic components in the Mandelbrot set. The method we use constitutes an extension for the complex domain of the one developed by Myrberg for the real map x ] x²−p, in which, given the symbolic sequence of a superstable orbit, the parameter value originating such a superstable orbit is worked out. We show that, when dealing with complex domain sequences, some of the solutions obtained correspond to the centres of the Mandelbrot sets hyperbolic components, while some others do not exist.     

Generating Fractals Using Geometric Algebra

In this paper we investigate how, using the language of Geometric Algebra [7, 4], the common escape-time Julia and Mandelbrot set fractals can be extended to arbitrary dimension and, uniquely, non-Eulidean geometries. We develop a geometric analog of complex numbers and show how existing ray-tracing techniques [2] can be extended. In addition, via the use of the Conformal Model for Geometric Algebra, we develop an analog of complex arithmetic for the Poincaré disc and show that, in non-Euclidean geometries, there are two related but distinct variants of the Julia and Mandelbrot sets.     

On the Julia set of the perturbed Mandelbrot map

In this work, we present numerical results which support the smooth decomposition method of the generalized Julia set by Peintge et al., in the case of other perturbations of the Mandelbrot map studied in our previous work (Argyris J, Andreadis I, Karakasidis T. Chaos, Solitons & Fractals 1999). We also calculate the generalized Julia set of a Mandelbrot map subject to noise. Hence, we are in a position to examining numerically the stability of this set under small noise.     

An effective algorithm to compute Mandelbrot sets in parameter planes

In McMullen (2000) it was proven that copies of generalized Mandelbrot set are dense in the bifurcation locus for generic families of rational maps. We develop an algorithm to an effective computation of the location and size of these generalized Mandelbrot sets in parameter space. We illustrate the effectiveness of the algorithm by applying it to concrete families of rational and entire maps.     

填充Jnlia集与Mandelbrot集的面积与直径

本文用单叶函数中的面积定理及Ｇａｒａｂｅｄｉａｎ－Ｓｃｈｉｆｆｅｒ不等式的有关推论．给出了求多项式的填充Ｊｕｌｉａ集及Ｍａｎｄｅｌｂｒｏｔ集面积的方法及直径的上界估计，从而给Ａ．Ｄｏｕａｄｙ所提的有关问题一个回答．