scholarly journals On the transition from regular to chaotic behaviors in the two degrees of freedom dynamical system

2007 ◽  
Vol 29 (3) ◽  
pp. 353-374
Author(s):  
Nguyen Van Khang ◽  
Nguyen Hoang Duong
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Dynamical Behavior ◽  
Control Parameters ◽  
Lagrange Equations ◽  
Chaotic Motions ◽  
Two Degrees Of Freedom

The main objective of the present paper is to study the transition from periodic regular mot ion to chaos in a two degrees of freedom dynamical system by changing control parameters. The nonlinear differential equations governing motion of the system are derived from the Lagrange equations. By use of the Poincare map, the dynamical behavior is identified based on numerical solutions of the ordinary differential equations. The Lyapunov exponent and the frequency spectrum are calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the considered system.

scholarly journals The interference galloping of two circular cylinders with equal diameters.

2007 ◽  
Vol 1 (1) ◽  
pp. 087-102
Author(s):  
Ewa Błazik-Borowa
Keyword(s):  
Differential Equations ◽  
Turbulence Intensity ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Numerical Analyses ◽  
Circular Cylinders ◽  
Two Degrees Of Freedom

The paper deals with numerical analyses of interference galloping of two elasticcaly supported circular cylinders of equal diameters. The basis of the analyses is a quasi-steady model of this phenomenon. The model assumes that both cylinders participate in the process of interference galloping and they have two degrees of freedom. The movement of the cylinders is described as a set of four nonlinear differential equations. On the basis of numerical solutions of these equations the author evaluate the correctness of this quasi-steady model. Then they estimate the dependence of a critical reduced velocity on the Scruton number, turbulence intensity and arrangements of the cylinders.

scholarly journals Dynamical analysis of a 2-degrees of freedom spatial pendulum

2018 ◽  
Vol 184 ◽  
pp. 01003 ◽  
Author(s):  
Stelian Alaci ◽  
Florina-Carmen Ciornei ◽  
Sorinel-Toderas Siretean ◽  
Mariana-Catalina Ciornei ◽  
Gabriel Andrei Ţibu
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Dynamical Analysis ◽  
Analysis Software ◽  
Lagrange Equations ◽  
Moments Of Inertia ◽  
Dynamical Simulation ◽  
Principal Moments Of Inertia

A spatial pendulum with the vertical immobile axis and horizontal mobile axis is studied and the differential equations of motion are obtained applying the method of Lagrange equations. The equations of motion were obtained for the general case; the only simplifying hypothesis consists in neglecting the principal moments of inertia about the axes normal to the oscillation axes. The system of nonlinear differential equations was numerically integrated. The correctness of the obtained solutions was corroborated to the dynamical simulation of the motion via dynamical analysis software. The perfect concordance between the two solutions proves the rightness of the equations obtained.

scholarly journals The Vibrations of a Particle about a Position of Equilibrium—Part 2 :The Relation between the Elliptic Function and Series Solutions

1921 ◽  
Vol 40 ◽  
pp. 34-49 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Elliptic Function ◽  
Degrees Of Freedom ◽  
Elliptic Functions ◽  
Series Solutions ◽  
Two Degrees Of Freedom ◽  
Function Solution

In a previous paper, entitled the “Vibrations of a Particle about a Position of Equilibrium,” by the author in collaboration with Professor E. B. Ross (Proc. Edin. Math. Soc., XXXIX, 1921, pp. 34–57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

scholarly journals Construction of Analytic Solution to Axisymmetric Flow and Heat Transfer on a Moving Cylinder

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1335
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stokes Equations ◽  
Nonlinear Differential Equations ◽  
Axisymmetric Flow ◽  
Control Parameters ◽  
Auxiliary Functions ◽  
Flow And Heat Transfer ◽  
Moving Cylinder ◽  
Convergence Control

Based on a new kind of analytical approach, namely the Optimal Auxiliary Functions Method (OAFM), a new analytical procedure is proposed to solve the problem of the annular axisymmetric stagnation flow and heat transfer on a moving cylinder with finite radius. As a novelty, explicit analytical solutions were obtained for the considered complex problem. First, the Navier–Stokes equations were simplified by means of similarity transformations that depended on different parameters and some combinations of these parameters, and the problem under study was reduced to six nonlinear ordinary differential equations with six unknowns. The OAFM proves to be a powerful tool for finding an accurate analytical solution for nonlinear problems, ensuring a fast convergence after the first iteration, even if the small or large parameters are absent, since the determination of the convergence-control parameters is independent of the magnitude of the coefficients that appear in the nonlinear differential equations. Concerning the main novelties of the proposed approach, it is worth mentioning the presence of some auxiliary functions, the involvement of the convergence-control parameters, the construction of the first iteration and much freedom to select the procedure for determining the optimal values of the convergence-control parameters.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

Finite Deflections of a Nonlinearly Elastic Bar

1970 ◽  
Vol 37 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J. T. Oden ◽  
S. B. Childs
Keyword(s):  
Differential Equations ◽  
Axial Force ◽  
Classical Theory ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Finite Deflections ◽  
Hyperbolic Tangent ◽  
Elastic Bar ◽  
Elastoplastic Materials ◽  
Nonlinearly Elastic

The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension of the classical theory of the elastica to include material nonlinearities. A moment-curvature relation in the form of a hyperbolic tangent law is introduced to simulate that of a class of elastoplastic materials. The problem of finite deflections of a clamped-end bar subjected to an axial force is given special attention, and numerical solutions to the resulting system of nonlinear differential equations are obtained. Tables of results for various values of the parameters defining the material are provided and solutions are compared with those of the classical theory of the elastica.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

scholarly journals VIRTUAL LABORATORIES AS STRATEGY FOR TEACHING IMPROVEMENT IN MATH SCIENCES AND ENGINEERING IN BOLIVIA

2020 ◽  
Vol 2 (1) ◽  
pp. 52-62
Author(s):  
Francisco Vargas
Keyword(s):  
Differential Equations ◽  
Matrix Algebra ◽  
Degrees Of Freedom ◽  
Inverted Pendulum ◽  
Numerical Algorithms ◽  
Three Dimensions ◽  
Technological Advancement ◽  
Teaching Improvement ◽  
Two Degrees Of Freedom ◽  
Virtual Laboratories

The vertiginous technological advancement has made necessary the use of computersoftware that contributes to the improvement of teaching in math sciences and engineering.It is in this context that the last five years the strategy presented in this article has been disseminatedin the main universities of Bolivia, a country where the schools have not yet been ableto offer basic disciplines such as calculus, matrix algebra, physics and/or differential equationsto solve problems considering applicative aspects. To establish this connection, it is necessaryto deduce differential equations associated with practical problems, solve these equationswith different numerical algorithms, and establish the concept of simulation to later introducelanguages like Python/VPython free of license to elaborate Virtual Laboratories that allow obtainingthe solutions in two and three dimensions. The classical problems addressed for thispurpose are the satellite of two degrees of freedom and the inverted pendulum.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

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