On the Local Convergence and Dynamics of New Iterative Method with Sixth Order Convergence

In this paper,we construct a new sixth order iterative method for solving nonlinear equations.The local convergence and order of convergence of the new iterative method is demonstrated.In order to check the validity of the new iterative method,we employ several chemical engineering applications and academic test problems.Numerical results show the good numerical performance of the new iterative method.Moreover,the dynamical study of the new method also supports the theoretical results.

Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link,and propose a new six-dimensional system,which exhibits some hidden attractors.The parameter switching algorithm is used to numerically study the dynamic behaviors of the system.Moreover,it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors.A self-excited attractor with the change of its parameters is also recognized.In addition,numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra,Lyapunov dimensions,bifurcation diagrams,phase space orbits,and basins of attraction.Consequently,the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable.These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems,and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper.Convergence analysis demonstrates that the local order of convergence of the numerical method is five.The computer algebra system CAS-Maple,Mathematica,or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects.Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms.A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior.Aside from visualizing iterative processes,this methodology provides useful information on iterations,such as the number of diverging-converging points and the average number of iterations as a function of initial points.Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance,efficiency,precision,and applicability of a newly presented technique.

Analysis of chaos behaviors of a bistable piezoelectric cantilever power generation system by the second-order Melnikov function

By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.

On Computer Implementation for Comparison of Inverse Numerical Schemes for Non-Linear Equations

In this research article,we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation simultaneously.These modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to 3.Convergence analysis proves that the orders of convergence of the two newly constructed inverse methods are 3.Using computer algebra system Mathematica,we find the lower bound of the convergence order and verify it theoretically.Dynamical planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB(R2011b),to present the global convergence properties of inverse simultaneous iterative methods as compared to classical methods.Some non-linear models are taken from Physics,Chemistry and engineering to demonstrate the performance and efficiency of the newly constructed methods.Computational CPU time,and residual graphs of the methods are provided to present the dominance behavior of our newly constructed methods as compared to existing inverse and classical simultaneous iterative methods in the literature.

Analysis and implementation of new fractional-order multi-scroll hidden attractors

To improve the complexity of chaotic signals,in this paper we first put forward a new three-dimensional quadratic fractional-order multi-scroll hidden chaotic system,then we use the Adomian decomposition algorithm to solve the proposed fractional-order chaotic system and obtain the chaotic phase diagrams of different orders,as well as the Lyaponov exponent spectrum,bifurcation diagram,and SE complexity of the 0.99-order system.In the process of analyzing the system,we find that the system possesses the dynamic behaviors of hidden attractors and hidden bifurcations.Next,we also propose a method of using the Lyapunov exponents to describe the basins of attraction of the chaotic system in the matlab environment for the first time,and obtain the basins of attraction under different order conditions.Finally,we construct an analog circuit system of the fractional-order chaotic system by using an equivalent circuit module of the fractional-order integral operators,thus realizing the 0.9-order multi-scroll hidden chaotic attractors.

Impact of radiation pressure and circumstellar dust on motion of a test particle in Manev’s field

In this paper,we present a study on the impact of radiation pressure and circumstellar dust on the motion of a test particle in the framework of the restricted four-body problem under the Manev’s field.We show that the distribution of equilibrium points on the plane of motion is slightly different from that of the classical Newtonian problem.With the aid of the Lyapunov characteristic exponents,we show that the system is sensitive to changes in initial conditions;hence,the orbit of the system is found to be chaotic in the phase space for the given initial conditions.Furthermore,a numerical application of this model to a stellar system(Gliese 667C)is considered,which validates the dependence of the equilibrium points on the mass parameter.We show that the non-collinear equilibrium points of this stellar system are distributed symmetrically about the x-axis,and five of them are linearly stable.The basins of attraction of the system show that the equilibrium points have irregular boundaries,and we use the energy integral and the Manev parameter to illustrate the zero-velocity curves showing the permissible region of motion of the test particle with respect to the Jacobi constant.

Complex dynamical behaviors in modulation-doped GaAs/Al_xGa_(1-x)As heterostructures

We study dynamics of the forced modulation-doped GaAs/AIGaAs heterostructure de-vices . The coupled differential equations governing the dynamical behaviors are numerically simulated. Biased with an appropriate dc field, the system exhibits two states: spontaneous current oscillation and fixed points. By imposing an ac driving force, the dynamical system shows frequency locking, quasiperiodicity, and chaos, which are sensitive to the amplitude and frequency of the externally ap-plied periodical microwave field. The basins of attraction of both ordinary attractors and strange attrac-tors are presented.