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

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 839 ◽  
Author(s):  
Young Hee Geum ◽  
Young Ik Kim
Keyword(s):  
Fixed Point ◽  
Iterative Methods ◽  
Parameter Space ◽  
Control Parameter ◽  
Elementary Theory ◽  
Linear Stability Theory ◽  
Bifurcation Points ◽  
Local Bifurcations ◽  
The Stability ◽  
Bifurcation Behavior

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.

LOCAL BIFURCATIONS OF A QUASIPERIODIC ORBIT

2012 ◽  
Vol 22 (12) ◽  
pp. 1250289 ◽  
Author(s):  
SOUMITRO BANERJEE ◽  
DAMIAN GIAOURIS ◽  
PETROS MISSAILIDIS ◽  
OTMAN IMRAYED
Keyword(s):  
Fixed Point ◽  
Unit Circle ◽  
Three Dimensional ◽  
Closed Curve ◽  
Invariant Curve ◽  
Poincaré Section ◽  
Poincare Section ◽  
Local Bifurcations ◽  
The Stability

We consider the local bifurcations that can occur in a quasiperiodic orbit in a three-dimensional map: (a) a torus doubling resulting in two disjoint loops, (b) a torus doubling resulting in a single closed curve with two loops, (c) the appearance of a third frequency, and (d) the birth of a stable torus and an unstable torus. We analyze these bifurcations in terms of the stability of the point at which the closed invariant curve intersects a "second Poincaré section". We show that these bifurcations can be classified depending on where the eigenvalues of this fixed point cross the unit circle.

STABILIZING A TWO-CELL DC-DC BUCK CONVERTER BY FIXED POINT INDUCED CONTROL

2009 ◽  
Vol 19 (06) ◽  
pp. 2043-2057 ◽  
Author(s):  
A. EL AROUDI ◽  
F. ANGULO ◽  
G. OLIVAR ◽  
B. G. M. ROBERT ◽  
M. FEKI
Keyword(s):  
Fixed Point ◽  
Numerical Simulations ◽  
Period Doubling ◽  
Buck Converter ◽  
Noisy Environment ◽  
Noise Levels ◽  
The Stability ◽  
Lyapunov Exponent Spectrum ◽  
Bifurcation Behavior ◽  
Power Electronic Converter

In this paper, we study nonlinear and bifurcation behavior of a two-cell DC-DC buck power electronic converter. The system shows nonsmooth period doubling bifurcation and chaotic phenomena in a certain zone of parameter space. This zone is located both analytically and from numerical simulations. One-dimensional, two-dimensional bifurcation diagrams and Lyapunov exponent spectrum are used to detect the different dynamic behaviors of the system. The Fixed Point Induced Control (FPIC) technique is applied to the system in order to widen the stability zone. The performance of the FPIC technique applied to the stabilization of a two-cell DC-DC buck converter is analyzed. With this technique, stabilization is achieved without changing the fixed point. The robustness in the presence of a noisy environment is checked by numerical simulations by considering different noise levels.

scholarly journals Grazing Orbits and Related Local Bifurcations of an Oscillator with Continuous and Piecewise-Linear Restoring Force

1996 ◽  
Vol 3 (1) ◽  
pp. 11-16
Author(s):  
Haiyan Hu
Keyword(s):  
Dynamic Systems ◽  
Numerical Scheme ◽  
Control Parameter ◽  
Critical Case ◽  
Piecewise Linear ◽  
Excitation Frequency ◽  
Restoring Force ◽  
Local Bifurcations ◽  
The Stability ◽  
Excited Oscillator

One critical case for the motion of a periodically excited oscillator with continuous and piecewise-linear restoring force is that the motion happens to graze a switching plane between two linear regions of the restoring force. This article presents a numerical scheme for locating the periodic grazing orbit first. Then, through a brief analysis, the article shows that the grazing phenomenon turns the stability trend of the periodic orbit so abruptly that it may be impossible to predict an incident local bifurcation with the variation of a control parameter from the concept of smooth dynamic systems. The numerical simulation in the article well supports the scheme and the analysis, and shows an abundance of grazing phenomena in an engineering range of the excitation frequency.

scholarly journals Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces

2003 ◽  
Vol 2003 (4) ◽  
pp. 193-216 ◽  
Author(s):  
Yakov Alber ◽  
Simeon Reich ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Fixed Point Problems ◽  
Proximal Methods ◽  
Approximation Methods ◽  
Nonexpansive Maps ◽  
The Stability ◽  
Weakly Contractive ◽  
Constraint Sets

We study descent-like approximation methods and proximal methods of the retraction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. We prove strong and weak convergences for weakly contractive and nonexpansive maps, respectively. We also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

scholarly journals Approximation theorems of a solution of amperometric enzymatic reactions based on Green’s fixed point normal-S iteration

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khanitin Muangchoo-in ◽  
Kanokwan Sitthithakerngkiet ◽  
Parinya Sa-Ngiamsunthorn ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Iterative Methods ◽  
Dynamical Problem ◽  
Enzymatic Reactions ◽  
Nonlinear Dynamical ◽  
Numerical Examples ◽  
Approximation Theorems ◽  
Kinetics Of ◽  
Nonlinear Dynamical Problem

AbstractIn this paper, the authors present a strategy based on fixed point iterative methods to solve a nonlinear dynamical problem in a form of Green’s function with boundary value problems. First, the authors construct the sequence named Green’s normal-S iteration to show that the sequence converges strongly to a fixed point, this sequence was constructed based on the kinetics of the amperometric enzyme problem. Finally, the authors show numerical examples to analyze the solution of that problem.

scholarly journals On the stability of set-valued functional equations with the fixed point alternative

2012 ◽  
Vol 2012 (1) ◽  
pp. 81 ◽  
Author(s):  
Hassan Kenary ◽  
Hamid Rezaei ◽  
Yousof Gheisari ◽  
Choonkil Park
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
The Stability

TERBIUM MOLYBDATE: GROUP THEORY AND FLUCTUATIONS

1987 ◽  
Vol 01 (05n06) ◽  
pp. 239-244
Author(s):  
SERGE GALAM
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Parameter Space ◽  
Landau Theory ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Theory Analysis ◽  
Continuous Transition ◽  
Dimensional Parameter ◽  
First Order

A new mechanism to explain the first order ferroelastic—ferroelectric transition in Terbium Molybdate (TMO) is presented. From group theory analysis it is shown that in the two-dimensional parameter space ordering along either an axis or a diagonal is forbidden. These symmetry-imposed singularities are found to make the unique stable fixed point not accessible for TMO. A continuous transition even if allowed within Landau theory is thus impossible once fluctuations are included. The TMO transition is therefore always first order. This explanation is supported by experimental results.

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