Abstract
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.
Highlights
A dynamical system can be formulated by any fixed rule describing the time-dependence of an evolving point with its position in the relevant state(phase)-space
An example of continuous dynamical systems can be seen in differential equations, while other examples of discrete dynamical systems can be seen in difference equations
This analysis will be limited to a discrete dynamical system which is governed by a difference equation in the form of an iterative method: with Ψ f as a fixed point operator [1]
Summary
A dynamical system can be formulated by any fixed rule describing the time-dependence of an evolving point with its position in the relevant state(phase)-space.
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