scholarly journals A New Reliable Numerical Method for Computing Chaotic Solutions of Dynamical Systems: The Chen Attractor Case

2015 ◽  
Vol 25 (13) ◽  
pp. 1550187 ◽  
Author(s):  
René Lozi ◽  
Alexander N. Pchelintsev
Keyword(s):  
Dynamical Systems ◽  
Power Series ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Chen System ◽  
Convergence Of Series ◽  
Special Cases ◽  
Backward Control ◽  
Numerical Precision ◽  
Region Of Convergence

This paper describes a new reliable numerical method for computing chaotic solutions of dynamical systems and, in special cases, is applied to Chen strange attractor. The numerical precision of the computation is finely mastered. We introduce a modification of the method of power series for the construction of approximate solutions of the Chen system together with forward/backward control of the precision. As a test for the method, we obtained the region of convergence of series and researched the behavior of the trajectories on this attractor. The results of a numerical experiment are presented.

ON THE NUMERICAL METHOD OF CONSTRUCTION OF UNSTABLE SOLUTIONS OF DYNAMICAL SYSTEMS WITH QUADRATIC NONLINEARITIES

2018 ◽  
pp. 555-565
Author(s):  
Alexander Nikolaevich Pchelintsev
Keyword(s):  
Dynamical Systems ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Convergence Of Series ◽  
Quadratic Nonlinearities ◽  
Unstable Solutions ◽  
Region Of Convergence

In this paper, the author considers the modification of the method of power series for the numerical construction of unstable solutions of systems of ordinary differential equations of chaotic type with quadratic nonlinearities in general form. A region of convergence of series is found and an algorithm for constructing approximate solutions is proposed.

scholarly journals Application of Piecewise Successive Linearization Method for the Solutions of the Chen Chaotic System

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
S. S. Motsa ◽  
Y. Khan ◽  
S. Shateyi
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Good Accuracy ◽  
Linearization Method ◽  
Chen System ◽  
Successive Linearization Method ◽  
Runge Kutta

This paper centres on the application of the new piecewise successive linearization method (PSLM) in solving the chaotic and nonchaotic Chen system. Numerical simulations are presented graphically and comparison is made between the PSLM and Runge-Kutta-based methods. The work shows that the proposed method provides good accuracy and can be easily extended to other dynamical systems including those that are chaotic in nature.

Implementation of DTM as a numerical study for the Casson fluid flow past an exponentially variable stretching sheet with thermal radiation

2021 ◽  
pp. 2150111
Author(s):  
Khadijah M. Abualnaja
Keyword(s):  
Fluid Flow ◽  
Thermal Radiation ◽  
Numerical Method ◽  
Stretching Sheet ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Transformation Method ◽  
Casson Fluid ◽  
Physical Parameters ◽  
Special Cases

This paper introduces a theoretical and numerical study for the problem of Casson fluid flow and heat transfer over an exponentially variable stretching sheet. Our contribution in this work can be observed in the presence of thermal radiation and the assumption of dependence of the fluid thermal conductivity on the heat. This physical problem is governed by a system of ordinary differential equations (ODEs), which is solved numerically by using the differential transformation method (DTM). This numerical method enables us to plot figures of the velocity and temperature distribution through the boundary layer region for different physical parameters. Apart from numerical solutions with the DTM, solutions to our proposed problem are also connected with studying the skin-friction coefficient. Estimates for the local Nusselt number are studied as well. The comparison of our numerical method with previously published results on similar special cases shows excellent agreement.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

Homotopy perturbation method for MHD non-Newtonian Williamson fluid over exponentially stretching sheet with viscous dissipation and convective boundary condition

2019 ◽  
Vol 30 (11) ◽  
pp. 1950088 ◽  
Author(s):  
Khadijah M. Abualnaja
Keyword(s):  
Fluid Flow ◽  
Perturbation Method ◽  
Numerical Method ◽  
Homotopy Perturbation Method ◽  
Stretching Sheet ◽  
Fluid Temperature ◽  
Williamson Fluid ◽  
Homotopy Perturbation ◽  
Exponential Stretching ◽  
Special Cases

This research is aimed at presenting the two-dimensional steady fluid flow, represented by Williamson constitutive model past a nonlinear exponential stretching sheet theoretically. The system of ODEs describing the physical problem is successfully solved numerically with the help of the homotopy perturbation method (HPM). Special attention is given to study the convergence analysis of the proposed method. The influences of the physical governing parameters acting on the fluid velocity and the fluid temperature are explained with the help of the figures and tables. Further, the presented numerical method is employed to calculate both the rate of heat transfer and the drag force for the Williamson fluid flow. In particular, it is observed that both the Eckert number and the dimensionless convective parameter have the effect of enhancing the temperature of the stretching surface, while the inverse was noted for the dimensionless mixed convection parameter. Finally, the comparison with previous numerical investigations of other authors at some special cases which is reported here proves that the results obtained via homotopy perturbation method are accurate and the numerical method is reliable.

A family of hyper-Bessel functions and convergent series in them

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Jordanka Paneva-Konovska
Keyword(s):  
Power Series ◽  
Bessel Function ◽  
Classical Theory ◽  
Bessel Functions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Convergent Series ◽  
Multi Index ◽  
Fatou Theorem ◽  
Convergence Of Series

AbstractThe Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

CHAOTIC ATTRACTORS OF THE CONJUGATE LORENZ-TYPE SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 3929-3949 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN ◽  
KUIFEI HUANG
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Chen System ◽  
Numerical Examples ◽  
Special Cases ◽  
Dynamical Properties

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions and the compound structure of the system are demonstrated with various numerical examples.

A Numerical Solution for the Hydrodynamic Lubrication of Finite Porous Journal Bearings

1973 ◽  
Vol 187 (1) ◽  
pp. 71-78 ◽  
Author(s):  
B. R. Reason ◽  
D. Dyer
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Journal Bearing ◽  
Hydrodynamic Lubrication ◽  
Approximate Solutions ◽  
Journal Bearings ◽  
Operating Conditions ◽  
Variable Porosity ◽  
Present Solution

We present a numerical solution for the operating conditions of a hydrodynamic porous journal bearing. The numerical method allows for the possibility of variable porosity in the bearing matrix, but the solution has been achieved on the assumption of matrix homogeneity. The relation between the various bearing parameters have been shown for a variety of bearing geometries and permeabilities enabling the operating conditions for this type of bearing to be better appreciated. A comparison of the present solution with approximate solutions used by other authors has been made, which indicates the useful working range of the approximate solutions.

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

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