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A study of the fractal graph generation of higher-order Julia sets based on a complex plane
Published in Artde Donald Kin-Tak Lam, Stephen D. Prior, Siu-Tsen Shen, Sheng-Joue Young, Liang-Wen Ji, Innovation in Design, Communication and Engineering, 2020
The Julia set is obtained by iterative computation using a complex function. Obviously, this set is a kind of fractal set. The mathematical expression of the quadratic complex function of a Julia set is as follows: F(z)=z2+c
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
Chaos refers to the apparently chaotic evolution of certain dynamical systems. Fractals are sets with an intriguing structure that reveals itself in shapes with intricate detail recurring on all scales. Julia sets are invariant sets for certain family of complex maps. Each Julia set corresponds to one map in the family. Typically the Julia sets are fractals and the dynamics of the corresponding map on the Julia set is chaotic.
Fractals and chaos
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. The Mandelbrot set can be thought of as an index for the Julia sets. This principle is exploited in virtually all results on the Mandelbrot set.
Wine and maths: mathematical solutions to wine–inspired problems
Published in International Journal of Mathematical Education in Science and Technology, 2018
As a real chaotic system, the fermentation of the grape must can be studied by means of biological mathematical models that explain the chemical phenomena influencing the production of wine. Recently, the sigmoidal logistic model has been used for simulations due to its property of good fit of experimental data. It is the solution of the following Cauchy problem: where p is the ethanol production, as described in [16]. Some dynamical system has a chaotic behaviour only in a subset of the phase space, namely on an attractor, since a large set of initial conditions will lead to orbits that converge to this chaotic region. Chaotic motion gives rise to what are known as strange attractors, that can have great detail and complexity such as the Lorentz attractor and fractals, whose most important representatives are the Mandelbrot set and the Julia set (see Figure 5).
Geometric limit of Julia set of a family of rational functions with odd degree
Published in Dynamical Systems, 2021
A. M. Alves, B. P. Silva e Silva, M. Salarinoghabi
Dynamics of a rational map on the Riemann sphere breaks into two pieces: dynamics on the Fatou set and dynamics on the Julia set. Fatou set of a map is ‘nicely behaved’ part of the dynamics, and it is completely understood through works of Fatou [7–9], Julia [11] and Sullivan [15]. The Julia set of a rational map is the ‘chaotic’ part of the dynamics. Substantial work by many people has been devoted to understanding these two sets for different families of rational maps.