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Buried Julia Components and Julia Sets
Qualitative Theory of Dynamical Systems, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Youming Wang +2 more
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Journal of Mathematical Sciences, 2020
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Julia sets converging to filled quadratic Julia sets
Ergodic Theory and Dynamical Systems, 2012AbstractIn this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled ...
Kozma, Robert T., Devaney, Robert L.
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Mathematics Magazine, 2015
Functions from the complex plane to itself are difficult to visualize; we consider the real and imaginary projections.
Julia A. Barnes +3 more
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Functions from the complex plane to itself are difficult to visualize; we consider the real and imaginary projections.
Julia A. Barnes +3 more
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The Mathematical Intelligencer, 1990
The author describes an algorithm for computing the Euclidean symmetry group of the Julia set of a given polynomial. It is based on his paper in Bull. Lond. Math. Soc. 22, No. 2, 576-582 (1990; Zbl 0725.30014).
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The author describes an algorithm for computing the Euclidean symmetry group of the Julia set of a given polynomial. It is based on his paper in Bull. Lond. Math. Soc. 22, No. 2, 576-582 (1990; Zbl 0725.30014).
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Chaos, Solitons & Fractals, 1992
Besides the exhaustive studies that have been made, of quadratic mappings on the real and complex fields, there has been some recent work with quaternions. Now the authors work in the 8-dimensional real algebra of octonions, and study the mapping \[ R^ a_ z:z \to z^ 2+c+u(a,c,z), \] where \(c\) and \(a\) are fixed, and \(u=c(za)-(cz)a\).
Griffin, C. J., Joshi, G. C.
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Besides the exhaustive studies that have been made, of quadratic mappings on the real and complex fields, there has been some recent work with quaternions. Now the authors work in the 8-dimensional real algebra of octonions, and study the mapping \[ R^ a_ z:z \to z^ 2+c+u(a,c,z), \] where \(c\) and \(a\) are fixed, and \(u=c(za)-(cz)a\).
Griffin, C. J., Joshi, G. C.
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Cayley’s problem and Julia sets
The Mathematical Intelligencer, 1984The goal of this exposition is to give a flavor of the subject of Julia Sets which we trace back to a problem posed by \textit{A. Cayley} [Am. J. Math. 2, 97 (1879)]. Our computer graphics not only illustrated the beauty that can be found in Julia sets, but they also provided us with insight that led us to some new results.
Peitgen, H. O. +2 more
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International Journal of Bifurcation and Chaos, 2008
The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z2 + c)eQ(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.
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The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z2 + c)eQ(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.
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