The Symmetry in the Noise-Perturbed Mandelbrot Set

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

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“…Some other recent results related to the Mandelbrot set can be found for example in [3][4][5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 89%

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

Trojovský

Venkatachalam

2021

Fractal Fract

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“…Mandelbrot [23] while analyzing the Julia set with different parameters for connectivity property discovered new fractal sets named after him also called M-set. Mandelbrot fractal set has been extensively used in many different areas recent examples of which can be found in [24], [25].…”

Section: ) Mandelbrot Setsmentioning

confidence: 99%

A New Chaotic Image Steganography Technique Based on Huffman Compression of Turkish Texts and Fractal Encryption With Post-Quantum Security

Kasapbaşi

2019

IEEE Access

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As the digital age is becoming an unavoidable part of our daily life people are becoming more concern about their privacy and security of their digital communications. Steganography and cryptography are the two most frequently used merits of digital life to fulfill these needs. In this study, a new spatial domain chaotic steganography scheme is presented utilizing also new fractal stream encryption algorithm to hide Huffman encoded and compressed Turkish texts. The study composes of four main phases. Firstly, a sample of Turkish newspaper columnist corpus is gathered to obtain not only the frequencies of letters including special Turkish characters but also punctuations, spaces, newlines, quotations, etc. As a result of the first phase, a static Huffman encoding dictionary is obtained for 102 encountered characters. Secondly, super Mandelbrot sets are used with the Logistic map to obtain one-time-pad stream keys to encrypt compressed texts. Thirdly, the LSB (Least Significant Bit) plain of the cover image is analyzed morphologically to find low entropy pixel locations so that in steganography phase spotted locations can be avoided and scheme can become more resistant against stego analysis. Lastly, the chaotic steganography phase is the final phase in which data hiding pixel is selected according to logistic map chaotic function then LSB of the selected pixel is updated. Many tests are carried out on many images to show the image quality namely PSNR, SNR, SSMI, UIQI, Entropy, MSE, and histogram. Results indicate that proposed schemes are successful for encryption and offer robust steganography. INDEX TERMS Chaotic steganography, fractal encryption, static Huffman.

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“…Moreover, the noiseperturbed generalized Mandelbrot sets were considered by [17], and by composing the additive and multiplicative noise, the perturbations of the generalized Mandelbrot set were also searched in [18]. With the aid of developing computer drawing tools, this realm of study has blossomed rapidly and attracted significant interest in recent years [19][20][21][22][23][24]. For instance, Wang et al presented a fractional Mandelbrot set [25], studied its dynamics in some detail [26], and then presented the impact of scale, memory, and impulse parameters on Mandelbrot sets and their fractal dimension [27].…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises

Popović,

Ersoy,

İnce

et al. 2024

Fractal Fract

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In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number’s fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature.

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The Symmetry in the Noise-Perturbed Mandelbrot Set

Cited by 3 publications

References 27 publications

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

A New Chaotic Image Steganography Technique Based on Huffman Compression of Turkish Texts and Fractal Encryption With Post-Quantum Security

Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises

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