Development of mandelbrot set for the logistic map with two parameters in the complex plane

In this paper, the study of the dynamical behavior of logistic map has been disused with representing fractals graphics of map, the logistic map depends on two parameters and works in the complex plane, the map defined by f(z,α,β)=αz(1–z)β. where and are complex numbers, and β is a positive integers number, the visualization method used in this work to generate fractals of the map and to inspect the relation between the value of β and the shape of the map, this visualization analysis showed also that, as the value of β increasing, as the number of humps in the function also increasing, and it demonstrate that is true also for the function’s first iteration , f2(x0)=f(f(x0)) and the second iteration , f3(x0)=f(f2(x0)), beside that , the visualization technique showed that the number of humps in that fractal is less than the ones in the second iteration of the original function ,the study of the critical points and their properties of the logistic map also discussed it, whereas finding the fixed point led to find the critical point of the function f, in addition , it haven proven for the set of all pointsα∈C and β∈N, the iteration function f(f(z) has an attractive fixed points, and belongs to the region specified by the disc |1–β(α–1)|<1. Also, The discussion of the Mandelbrot set of the function defined by the f(f(z)) examined in complex plans using the path principle, such that the path of the critical point z=z0 is restricted, finally, it has proven that the Mandelbrot set f(z,α,β) contains all the attractive fixed points and all the complex numbers in which α≤(1/β+1) (1/β+1) and the region containing the attractive fixed points for f2(z,α,β) was identified

A New Family of High-Order Ehrlich-Type Iterative Methods

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Two Classes of Iteration Functions and Q-Convergence of Two Iterative Methods for Polynomial Zeros

In this work, two broad classes of iteration functions in n-dimensional vector spaces are introduced. They are called iteration functions of the first and second kind at a fixed point of the corresponding iteration function. Two general local convergence theorems are presented for Picard-type iterative methods with high Q-order of convergence. In particular, it is shown that if an iterative method is generated by an iteration function of first or second kind, then it is Q-convergent under each initial approximation that is sufficiently close to the fixed point. As an application, a detailed local convergence analysis of two fourth-order iterative methods is provided for finding all zeros of a polynomial simultaneously. The new results improve the previous ones for these methods in several directions.

A Clustering Perspective of the Collatz Conjecture

This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to the visualization of behaviors of the Collatz iteration function and the analysis of the results.

The MULTIPLISITAS NEWTON DAN TITIK TETAP ATRAKTIF DALAM MENENTUKAN KEKONVERGENAN

Newton's method is one of the numerical methods used in finding polynomial roots. This method will be very effective to use, if the initial estimate of the roots for the Newton iteration function satisfies sufficient Newtonian convergence, [11]. In this article we will analyze the efficacy of this method by looking at the relationship between the fixed point method and Newton's iteration function. When the iteration of the function converges to the root, the velocity of convergence can also be determined. In terms of the speed of convergence, it turns out to be very dependent on the multiplicity of Newton's method itself.

Regenerasi Fungsi x2-9x-99 Dalam Pembangkit Bilangan Acak Berbasis CSRPNG Chaos

This study examines whether the function f(x)=x2-9x-99 can be used as a key generator in cryptography. The quadratic function is regenerated using the fixed point iteration method into an iteration function. The distribution of digits to the output of iterative function to generate a number of chaos. Randomization testing uses run test and monobit testing. Followed by cryptographic testing to get the correlation between ciphertext and key which will be used as a decision whether the resulting key is random or not. Based on research that has been done iteration function xi = (xi-12-9xi-1-99)/9 can generate CSRPNG Chaos random numbers with the correlation level closest to the value of 0.

Julia Set with Trigonometric Function

Julia sets are generated by initializing a complex number z = x + yi where z is then iterated using the iteration function fc (z)= zn 2 + c, where n indicates the number of iteration and c is a constant complex number. Recently, study of cubic Julia sets was introduced in Noor Orbit (NO) with improved escape criterions for cubic polynomials. In this paper, we investigate the complex dynamics of different functions and apply the iteration function to generate an entire new class of Julia sets. Here, we introduce different types of orbits on cubic Julia sets with trigonometric functions. The two functions to investigate from Julia sets are sine and cosine functions.