scholarly journals Non-Differentiable Structure in the Generalized Mandelbrot Set

1988 ◽  
Vol 43 (3) ◽  
pp. 287-288
Author(s):  
J. Peinke ◽  
J. Parisi ◽  
B. Röhricht ◽  
O. E. Rössler ◽  
W. Metzler
Keyword(s):  
Vortex Structure ◽  
Julia Set ◽  
Mandelbrot Set ◽  
Fractal Structures ◽  
Differentiable Structure ◽  
Self Similar

Abstract The generalized Mandelbrot set, described previously, contains - like the original Mandelbrot set - non-differentiable self-similar fractal structures. An example looking like a vortex structure is presented along with a corresponding generalized Julia set.

scholarly journals Fractal structures in freezing brine

2017 ◽  
Vol 826 ◽  
pp. 975-995
Author(s):  
Sergey Alyaev ◽  
Eirik Keilegavlen ◽  
Jan Martin Nordbotten ◽  
Iuliu Sorin Pop
Keyword(s):  
Characteristic Length ◽  
Complex Problem ◽  
Rate Of Change ◽  
Critical Threshold ◽  
Fractal Structures ◽  
Primary Interest ◽  
External Temperature ◽  
Self Similar ◽  
Salt Diffusion ◽  
Heuristic Analysis

The process of initial ice formation in brine is a highly complex problem. In this paper, we propose a mathematical model that captures the dynamics of nucleation and development of ice inclusions in brine. The primary emphasis is on the interaction between ice growth and salt diffusion, subject to external forcing provided by temperature. Within this setting two freezing regimes are identified, depending on the rate of change of the temperature: a slow freezing regime where a continuous ice domain is formed; and a fast freezing regime where recurrent nucleation appears within the fluid domain. The second regime is of primary interest, as it leads to fractal-like ice structures. We analyse the critical threshold between the slow and fast regimes by identifying the explicit rates of external temperature control that lead to self-similar salt-concentration profiles in the fluid domains. Subsequent heuristic analysis provides estimates of the characteristic length scales of the fluid domains depending on the time-variation of the temperature. The analysis is confirmed by numerical simulations.

scholarly journals Wavelet introscopy of human organism bionets

2021 ◽  
pp. 63-72
Author(s):  
Gennady M. Aldonin ◽  
◽  
Vasily V. Cherepanov ◽  
Keyword(s):  
Functional State ◽  
Human Body ◽  
Dynamic Models ◽  
Golden Ratio ◽  
The State ◽  
The Body ◽  
Human Organism ◽  
Fractal Structures ◽  
Self Similar ◽  
The Creation

In domestic and foreign practice, a great deal of experience has been accumulated in the creation of means for monitoring the functional state of the human body. The existing complexes mainly analyze the electrocardiogram, blood pressure and a number of other physiological parameters. Diagnostics is often based on formal statistical data which are not always correct due to the nonstationarity of bioprocesses and without taking into account their physical nature. An urgent task of monitoring the state of the cardiovascular system is the creation of effective algorithms for computer technologies to process biosignals based on nonlinear dynamic models of body systems since biosystems and bioprocesses have a nonlinear nature and fractal structure. The nervous and muscular systems of the heart, the vascular and bronchial systems of the human body are examples of such structures. The connection of body systems with their organization in the form of self-similar fractal structures with scaling close to the “golden ratio” makes it possible to diagnose them topically. It is possible to obtain detailed information about the state of the human body’s bio-networks for topical diagnostics on the basis of the wavelet analysis of biosignals (the so-called wavelet-introscopy). With the help of wavelet transform, it is possible to reveal the structure of biosystems and bioprocesses, as a picture of the lines of local extrema of wavelet diagrams of biosignals. Mathematical models and software for wavelet introscopy make it possible to extract additional information from biosignals about the state of biosystems. Early detection of latent forms of diseases using wavelet introscopy can shorten the cure time and reduce the consequences of disorders of the functional state of the body (FSO), and reduce the risk of disability. Taking into account the factors of organizing the body’s biosystems in the form of self-similar fractal structures with a scaling close to the “golden ratio” makes it possible to create a technique for topical diagnostics of the most important biosystems of the human body.

LIMITING DYNAMICS FOR THE COMPLEX STANDARD FAMILY

1995 ◽  
Vol 05 (03) ◽  
pp. 673-699 ◽  
Author(s):  
NÚRIA FAGELLA
Keyword(s):  
Cross Sections ◽  
Julia Set ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Dimensional Parameter ◽  
Quadratic Polynomials ◽  
New Family ◽  
Standard Family ◽  
The Family ◽  
Characteristic Features

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

scholarly journals Multicorns are not path connected

2014 ◽  
Author(s):  
John Hamal Hubbard ◽  
Dierk Schleicher
Keyword(s):  
Julia Set ◽  
Mandelbrot Set ◽  
Quadratic Polynomials ◽  
Unicritical Polynomials ◽  
Path Connected ◽  
Special Case ◽  
Connectedness Locus

This chapter proves that the tricorn is not locally connected and not even pathwise connected, confirming an observation of John Milnor from 1992. The tricorn is the connectedness locus in the space of antiholomorphic quadratic polynomials z ↦ ̄z² + c. The chapter extends this discussion more generally for antiholomorphic unicritical polynomials of degrees d ≥ 2 and their connectedness loci, known as multicorns. The multicorn M*subscript d is the connectedness locus in the space of antiholomorphic unicritical polynomials psubscript c(z) = ̄zsubscript d + c of degree d, i.e., the set of parameters for which the Julia set is connected. The special case d = 2 is the tricorn, which is the formal antiholomorphic analog to the Mandelbrot set.

Shadow of a disformal Kerr black hole in quadratic degenerate higher-order scalar–tensor theories

2020 ◽  
Vol 80 (12) ◽  
Author(s):  
Fen Long ◽  
Songbai Chen ◽  
Mingzhi Wang ◽  
Jiliang Jing
Keyword(s):  
Black Hole ◽  
Deformation Parameter ◽  
Kerr Black Hole ◽  
Arbitrary Spin ◽  
Higher Order ◽  
Fractal Structures ◽  
Extreme Black Hole ◽  
Spin Parameter ◽  
Result Show ◽  
Self Similar

AbstractWe have studied the shadow of a disformal Kerr black hole with an extra deformation parameter, which belongs to non-stealth rotating solutions in quadratic degenerate higher-order scalar–tensor (DHOST) theory. Our result show that the size of the shadow increases with the deformation parameter for the black hole with arbitrary spin parameter. However, the effect of the deformation parameter on the shadow shape depends heavily on the spin parameter of black hole and the sign of the deformation parameter. The change of the shadow shape becomes more distinct for the black hole with the more quickly rotation and the more negative deformation parameter. Especially, for the near-extreme black hole with negative deformation parameter, there exist a “pedicel”-like structure appeared in the shadow, which increases with the absolute value of deformation parameter. The eyebrow-like shadow and the self-similar fractal structures also appear in the shadow for the disformal Kerr black hole in DHOST theory. These features in the black hole shadow originating from the scalar field could help us to understand the non-stealth disformal Kerr black hole and quadratic DHOST theory.

Projected Measures: A Simple Way to Characterize Fractal Structures and Interfaces

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 295-308
Author(s):  
Massimiliano Giona ◽  
Manuela Giustiniani ◽  
Oreste Patierno
Keyword(s):  
Closed Form ◽  
Fourier Transforms ◽  
Fractal Structures ◽  
Multifractal Spectra ◽  
Self Similar ◽  
The Moment ◽  
Dynamic Phenomena

The properties of projected measures of fractal objects are investigated in detail. In general, projected measures display multifractal features which play a role in the evolution of dynamic phenomena on/through fractal structures. Closed-form results are obtained for the moment hierarchy of model fractal interfaces. The distinction between self-similar and self-affine interfaces is discussed by considering the properties of multifractal spectra, the orientational effects in the behavior of the moment hierarchies, and the scaling of the corresponding Fourier transforms. The implications of the properties of projected measures in the characterization of transfer phenomena across fractal interfaces are briefly analyzed.

Conformal trace theorem for Julia sets of quadratic polynomials

2017 ◽  
Vol 39 (9) ◽  
pp. 2481-2506 ◽  
Author(s):  
A. CONNES ◽  
E. MCDONALD ◽  
F. SUKOCHEV ◽  
D. ZANIN
Keyword(s):  
Hausdorff Dimension ◽  
Jordan Curve ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Trace Theorem ◽  
Quadratic Polynomials ◽  
Full Proof

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

scholarly journals Construction of 3D Mandelbrot Set and Julia Set

2014 ◽  
Vol 85 (15) ◽  
pp. 32-36
Author(s):  
Ashish Negi ◽  
Ankit Garg ◽  
Akshat Agrawal
Keyword(s):  
Julia Set ◽  
Mandelbrot Set
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