Research on Chaotic Orbit in Mandelbrot Set

2014 ◽  
Vol 912-914 ◽  
pp. 1349-1352
Author(s):  
Tao Sui ◽  
Xiao Yu Zhang ◽  
Guang Shen Li ◽  
Guan Nan Liu
Keyword(s):  
Recurrence Formula ◽  
Periodic Points ◽  
Mandelbrot Set ◽  
Chaotic Orbit ◽  
Java Applet ◽  
Distribution Rules ◽  
Period Orbit ◽  
Fractal Images ◽  
Relationship Of ◽  
Stable Points

In this paper, the chaotic orbit in Mandelbrot set is introduced. On the basis of other scholars research, the character and distribution rules of pre-periodic orbits and pre-periodic points-Misiurewiz points about Mandelbrot set chaos-fractal images were studied. The software of constructing the general M-J set with Java Applet is improved. Using the method of computer mathematic experiments, the paper analyses the fixed orbit and period orbit in the M-set, gains the topology relationship of M set in super stable points, a recurrence formula between the period orbit and M-set periodic-buds is created.

Mobile Fractal Generation

2011 ◽  
pp. 399-413 ◽  
Author(s):  
Daniel C. Doolan ◽  
Sabin Tabirca ◽  
Laurence T. Yang
Keyword(s):  
Mobile Phone ◽  
Ubiquitous Computing ◽  
Mobile Devices ◽  
Mobile Phones ◽  
New Age ◽  
Mandelbrot Set ◽  
Processing Task ◽  
The Past ◽  
Fractal Images ◽  
Mobile Phone Usage

Ever since the discovery of the Mandelbrot set, the use of computers to visualise fractal images have been an essential component. We are looking at the dawn of a new age, the age of ubiquitous computing. With many countries having near 100% mobile phone usage, there is clearly a potentially huge computation resource becoming available. In the past years there have been a few applications developed to generate fractal images on mobile phones. This chapter discusses three possible methodologies whereby such images can be visualised on mobile devices. These methods include: the generation of an image on a phone, the use of a server to generate the image and finally the use of a network of phones to distribute the processing task.

CHAOS AND FRACTAL OF THE GENERAL TWO-DIMENSIONAL QUADRATIC MAP

2008 ◽  
Vol 22 (20) ◽  
pp. 3461-3471
Author(s):  
XINGYUAN WANG
Keyword(s):  
Fractal Dimension ◽  
General Feature ◽  
Strange Attractors ◽  
Two Dimensional ◽  
Control Parameters ◽  
Self Similarity ◽  
Boundary Equation ◽  
Quadratic Map ◽  
Fractal Images ◽  
Stable Points

The nature of the stable points of the general two-dimensional quadratic map is considered analytically, and the boundary equation of the first bifurcation of the map in the parameter space is given out. The general feature of the nonlinear dynamic activities of the map is analyzed by the method of numerical computation. By utilizing the Lyapunov exponent as a criterion, this paper constructs the strange attractors of the general two-dimensional quadratic map, and calculates the fractal dimension of the strange attractors according to the Lyapunov exponents. At the same time, the researches on the fractal images of the general two-dimensional quadratic map make it clear that when the control parameters are different, the fractal images are different from each other, and these fractal images exhibit the fractal property of self-similarity.

Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

2016 ◽  
Vol 26 (01) ◽  
pp. 1630002 ◽  
Author(s):  
M. Fernández-Guasti
Keyword(s):  
Parameter Space ◽  
Complex Structure ◽  
Three Dimensional ◽  
Periodic Points ◽  
Mandelbrot Set ◽  
Three Dimensions ◽  
Quadratic Mapping ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Nilpotent Elements

The quadratic iteration is mapped within a nondistributive imaginary scator algebra in [Formula: see text] dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period [Formula: see text], are shown to be necessarily surrounded by points that produce a divergent magnitude after [Formula: see text] iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the [Formula: see text]th iteration, have preperiod 1 and period [Formula: see text]. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.

Fractal diffusion patterns of periodic points in the Mandelbrot set

2021 ◽  
Vol 153 ◽  
pp. 111599
Author(s):  
Dakuan Yu ◽  
Wurui Ta ◽  
Youhe Zhou
Keyword(s):  
Periodic Points ◽  
Mandelbrot Set

scholarly journals Generation of 3D Fractal Images for Mandelbrot and Julia Sets

2011 ◽  
pp. 14-18
Author(s):  
Bulusu Rama ◽  
Jibitesh Mishra
Keyword(s):  
Real World ◽  
Julia Set ◽  
Computational Modelling ◽  
Mandelbrot Set ◽  
Self Similarity ◽  
3D Images ◽  
Scale Down ◽  
Fractal Images ◽  
Similarity Scale ◽  
Fractal Landscapes

Fractals provide an innovative method for generating 3D images of real-world objects by using computational modelling algorithms based on the imperatives of self-similarity, scale invariance, and dimensionality. Images such as coastlines, terrains, cloud mountains, and most interestingly, random shapes composed of curves, sets of curves, etc. present a multivaried spectrum of fractals usage in domains ranging from multi-coloured, multi-patterned fractal landscapes of natural geographic entities, image compression to even modelling of molecular ecosystems. Fractal geometry provides a basis for modelling the infinite detail found in nature. Fractals contain their scale down, rotate and skew replicas embedded in them. Many different types of fractals have come into limelight since their origin. This paper explains the generation of two famous types of fractals, namely the Mandelbrot Set and Julia Set, the3D rendering of which gives a real-world look and feel in the world of fractal images.

scholarly journals Parametric 2-dimensional L systems and recursive fractal images: Mandelbrot set, Julia sets and biomorphs

2002 ◽  
Vol 26 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Alfonso Ortega ◽  
Marina de la Cruz ◽  
Manuel Alfonseca
Keyword(s):  
Julia Sets ◽  
Mandelbrot Set ◽  
Fractal Images ◽  
L Systems

scholarly journals FAMILY OF INVARIANT CANTOR SETS AS ORBITS OF DIFFERENTIAL EQUATIONS II: JULIA SETS

2011 ◽  
Vol 21 (01) ◽  
pp. 77-99 ◽  
Author(s):  
YI-CHIUAN CHEN ◽  
TOMOKI KAWAHIRA ◽  
HUA-LUN LI ◽  
JUAN-MING YUAN
Keyword(s):  
Differential Equations ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Continuous Family ◽  
Coupled Differential Equations ◽  
External Rays ◽  
Period Orbit ◽  
External Angle ◽  
Integral Curves

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.

The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽  
1980 ◽  
Vol 6 (02) ◽  
pp. 146-160 ◽  
Author(s):  
William A. Oliver
Keyword(s):  
Critical Points ◽  
Scleractinian Corals ◽  
Convincing Evidence ◽  
Early Triassic ◽  
Rugose Corals ◽  
History Of ◽  
The Subject ◽  
Relationship Of ◽  
The Relationship ◽  
Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

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