Conjugacy Map Research Articles

On a Sixth-order Solver for Multiple Roots of Nonlinear Equations

A number of iterative schemes with high convergence order to solve nonlinear equations are presented in the literature. In this paper, a sixth-order multiple-zero finder has been developed and the dynamics of selected iterative schemes with uniparametric polynomial weight function are investigated using M\"{o}bius conjugacy map applied to the form $((z-A)(z-B))^m$. The complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points are studied and the numerical experiments are carried out.

On a Parameter-controlled Method for Multiple Zeros

For a parameter-controlled Newton-secant method, we investigate the complex dynamics with the stability surfaces and parameter spaces using Möbious conjugacy map applied to the form $((z-A) (z-B))^m$. The basins of attraction of test functions is illustrated according to the color palette. The nonlinear equation related to blood rheology model has been employed to show the basins of attraction.

A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

On the ring M n ( C ) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the “properness” property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the “ * -order” on M n ( C ) . More generally, related questions are asked and answered about the ring M n ( R ) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding “ * -transposition” involution on M n ( R ) . A criterion is found for this involution of M n ( R ) to be proper, so that M n ( R ) has a corresponding * -order. The special case R = Z k is also considered.

A Family of Optimal Cubic-Order Multiple-root Solvers and Their Dynamics

The complex dynamical analysis of the cubic-order iterative family is proposed to draw the fractal images via M\"{o}bius conjugacy map applied to a quadratic polynomial $(z-A) (z-B)$. The resulting dynamics is clearly visualized through various stability surfaces and parameter spaces using Mathematica.

Computational Bifurcations Occurring on Red Fixed Components in the λ-Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map

Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The λ -parameters for periodic orbits in the parameter plane are painted in different colors depending on their periods, and the bounded or chaotic ones are colored black to illustrate λ -dependent connected components. When a red fixed component in the parameter plane branches into a q-periodic component, we encounter geometric bifurcation phenomena whose characteristics determine the desired boundary equation and bifurcation point. Computational results along with illustrated components support the bifurcation phenomena underlying this paper.

Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders

An analysis based on an elementary theory of plane curves is presented to locate bifurcation points from a main component in the parameter space of a family of optimal fourth-order multiple-root finders. We explore the basic dynamics of the iterative multiple-root finders under the Möbius conjugacy map on the Riemann sphere. A linear stability theory on local bifurcations is developed from the viewpoint of an arbitrarily small perturbation about the fixed point of the iterative map with a control parameter. Invariant conjugacy properties are established for the fixed point and its multiplier. The parameter spaces and dynamical planes are investigated to analyze the underlying dynamics behind the iterative map. Numerical experiments support the theory of locating bifurcation points of satellite and primitive components in the parameter space.

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper.

On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the Möbius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map.

The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere

In this paper, we present dynamical viewpoints under the Mobius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root finders. Various conjugacy properties are investigated including the invariance of the fixed point and its multiplier, which enables the family of iterative methods to trace along the consistent orbits in position. The parameter spaces and dynamical planes are studied and illustrated to visualize the periodic components and their geometric properties. Both theoretical and computational analyses along with a numerical algorithm are carried out regarding the bifurcation points of satellite and primitive components.

A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions

A generic family of sixth-order modified Newton-like multiple-zero finders have been proposed in Geum et al. (2016). Among them we select a specific family of iterative methods with uniparametric bivariate polynomial weight functions and study their dynamics by investigating the relevant parameter spaces and dynamical planes, via Möbius conjugacy map applied to a prototype polynomial of the form ( z − a ) m ( z − b ) m . The resulting dynamics is best illustrated through a variety of parameter spaces as well as dynamical planes.

Dynamical Analysis via Möbius Conjugacy Map on a Uniparametric Family of Optimal Fourth-Order Multiple-Zero Solvers with Rational Weight Functions

A triparametric family of fourth-order multiple-zero solvers have been proposed. In this paper, we select among them a uniparametric family of optimal fourth-order multiple-zero solvers with rational weight functions and pursue their dynamics by exploring the relevant parameter spaces and dynamical planes, by means of Möbius conjugacy map applied to a prototype polynomial of the form (z-A)m(z-B)m. The resulting dynamics is best illustrated through various stability surfaces and parameter spaces as well as dynamical planes.

On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio

Under the assumption of known root multiplicity m∈N, a triparametric family of two-point optimal quartic-order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function-to-function ratio. Special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper. In addition, we explore the relevant dynamics of proposed methods via Möbius conjugacy map when applied to a prototype polynomial (z−a)m(z−b)m. The results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes.

White Gaussian Chaos

A discrete-time white Gaussian noise (WGN) is a random process with impulsive autocorrelation function. Also, the random variables obtained by sampling the process at any time instants are jointly Gaussian. WGN is widely used to model noise in engineering and physics. In this letter, we propose a way to generate chaotic signals that behave like WGN, due to the features of its autocorrelation function and its invariant density. From the tent map and by using as conjugacy map transformations commonly employed to random variables, we obtain a white Gaussian chaos (WGC) map. Numerical simulations are shown to illustrate the technique. WGC can be an interesting choice for chaos generator in chaos-based communication systems.

The Collatz conjecture and De Bruijn graphs

We study variants of the well-known Collatz graph, by considering the action of the 3n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz graph, these graphs are very structured, and have several interesting properties. We then look at a natural generalization of these finite graphs to the 2-adic integers, and show that the isomorphism between the resulting infinite graphs is exactly the conjugacy map previously studied by Bernstein and Lagarias. Finally, we show that for generalizations of the 3n+1 function, such as the family of an+b functions and Collatz-like functions, we get similar relations with 2-adic and p-adic De Bruijn graphs respectively.

Non-standard smooth realizations of Liouville rotations

AbstractWe augment the $C^\infty $ conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

Discretization of circle maps

To investigate the dynamical behavior of a discrete dynamical system given by a map f, it is nowadays a standard method to look at the discretization of the Frobenius-Perron operator w.r.t. to a box-partition of the state space resulting in a transition matrix M(f). The aim is to obtain characteristics of f by that of M(f) - e.g. invariant measures and their densities.¶In this paper we will treat the special case of circle maps \(f\colon S^1\rightarrow S^1\) which we assume to be orientation preserving C2-diffeomorphisms. They appear in several applications as for instance in investigations of the dynamic on invariant curves or tori.¶We will find a number of relations between f and M(f) concerning the rotation number and the ergodic measure of f presented by the Denjoy conjugacy map. Approximations for the rotation number \(\rho(f)\) based on M(f) and an a posteriori error estimation for the invariant measure of ergodic circle maps given by the Frobenius eigenvector of M(f) will be constructed.