相关论文: Higher Order Mandelbrot Sets and their Varying Sha…

Direct look at the celebrated "chaotic" Mandelbrot Set in Fig..\ref{Mand2} immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific {\it forest} structure. In /hep-th/9501235 a systematic…

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We…

In this paper the fractional-order Mandelbrot and Julia sets in the sense of $q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are introduced and several properties are analytically and numerically studied. Some…

In this paper it is shown analytically and computationally that the Mandelbrot set of integer order are particular cases of Julia sets of Caputo s like fractional order. Also the differences between the fractional-order Mandelbrot set and…

A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating…

We describe an interesting interplay between symbolic dynamics, the structure of the Mandelbrot set, permutations of periodic points achieved by analytic continuation, and Galois groups of certain polynomials. Internal addresses are a…

B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is…

Mandelbrot set is a closure of the set of zeroes of $resultant_x(F_n,F_m)$ for iterated maps $F_n(x)=f^{\circ n}(x)-x$ in the moduli space of maps $f(x)$. The wonderful fact is that for a given $n$ all zeroes are not chaotically scattered…

We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification…

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers.…

The generation of fractals and study of the dynamics of polynomials is one of the emerging and interesting field of research nowadays. We introduce in this paper the dynamics of polynomials z^ n - z + c = 0 for n>=2 and applied Jungck…

We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit…

The Mandelbrot set is an extremely well-known mathematical object that can be described in a quite simple way but has very interesting and non-trivial properties. This paper surveys some results that are known concerning the…

We investigate with the help of Clifford algebraic methods the Mandelbrot set over arbitrary two-component number systems. The complex numbers are regarded as operator spinors in D\times spin(2) resp. spin(2). The thereby induced (pseudo)…

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, add the…

The so-called "pinched disk" model of the Mandelbrot set is due to A.~Douady, J.~H.~Hubbard and W.~P.~Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk…

We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the…

Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…

We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into…

In this article, we give the exact interval of the cross section of the so called Mandelbric set generated by the polynomial $z^3+c$ where $z$ and $c$ are complex numbers. Following that result, we show that the Mandelbric defined on the…