scholarly journals An Iteration Scheme Suitable for Solving Limit Cycles of Nonsmooth Dynamical Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Q. X. Liu ◽  
Y. M. Chen ◽  
J. K. Liu
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Nonlinear Damping ◽  
Iteration Scheme ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Nonsmooth Systems ◽  
Nonlinear Algebraic Equations ◽  
Nonsmooth Dynamical Systems ◽  
Previous Iteration

The Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities and/or nonlinear damping terms. If the MIM is implemented routinely, the unknown frequency and amplitude of limit cycle (LC) would couple to each other in complicated nonlinear algebraic equations at each iteration. It is cumbersome to solve these algebraic equations, especially for nonsmooth systems. In the modified procedures, the unknown frequency is substituted by the determined value obtained at the previous iteration. By this means, the frequency is decoupled from the nonlinear terms. Numerical examples show that the LCs obtained by the modified MIM agree well with numerical results. The presented method is very suitable for solving self-excited systems, especially those with nonlinear damping and nonsmooth nonlinearities.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

A GLOBAL PERIOD-1 MOTION OF A PERIODICALLY EXCITED, PIECEWISE-LINEAR SYSTEM

2005 ◽  
Vol 15 (06) ◽  
pp. 1945-1957 ◽  
Author(s):  
SANTHOSH MENON ◽  
ALBERT C. J. LUO
Keyword(s):  
Linear System ◽  
Numerical Simulations ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Piecewise Linear System ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Nonsmooth Dynamical Systems

The period-1 motion of a piecewise-linear system under a periodic excitation is predicted analytically through the Poincaré mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonlinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the piecewise-linear system and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical systems.

scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. H. Behiry
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Present Method ◽  
Bernstein Polynomials ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Nonlinear Algebraic Equations

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

scholarly journals Using chaos synchronization to estimate the largest lyapunov exponent of nonsmooth systems

2000 ◽  
Vol 4 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Andrzej Stefanski ◽  
Tomasz kapitaniak
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dry Friction ◽  
Chaos Synchronization ◽  
Duffing Oscillator ◽  
Largest Lyapunov Exponent ◽  
Nonsmooth Systems ◽  
The Largest Lyapunov Exponent ◽  
Nonsmooth Dynamical Systems

We describe the method of estimation of the largest Lyapunov exponent of nonsmooth dynamical systems using the properties of chaos synchronization. The method is based on the coupling of two identical dynamical systems and is tested on two examples of Duffing oscillator: (i) with added dry friction, (ii) with impacts.

scholarly journals Bernoulli Wavelets Operational Matrices Method for the Solution of Nonlinear Stochastic Itô-Volterra Integral Equations

2020 ◽  
pp. 395-410
Author(s):  
S. C. Shiralashetti ◽  
Lata Lamani
Keyword(s):  
Integral Equations ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Operational Matrices ◽  
Effective Strategy ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Stochastic Operational Matrix ◽  
Nonlinear Algebraic Equations ◽  
Operational Matrix Of Integration

This article gives an effective strategy to solve nonlinear stochastic Itô-Volterra integral equations (NSIVIE). These equations can be reduced to a system of nonlinear algebraic equations with unknown coefficients, using Bernoulli wavelets, their operational matrix of integration (OMI), stochastic operational matrix of integration (SOMI) and these equations can be solved numerically. Error analysis of the proposed method is given. Moreover, the results obtained are compared to exact solutions with numerical examples to show that the method described is accurate and precise.

scholarly journals A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
C. Guler ◽  
S. O. Kaya
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Numerical Approach ◽  
The Other ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Taylor Coefficients ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Taylor Polynomials

In this study, a matrix method based on Taylor polynomials and collocation points is presented for the approximate solution of a class of nonlinear differential equations, which have many applications in mathematics, physics and engineering. By means of matrix forms of the Taylor polynomials and their derivatives, the technique we have used reduces the solution of the nonlinear equation with mixed conditions to the solution of a matrix equation which corresponds to a system of nonlinear algebraic equations with the unknown Taylor coefficients. On the other hand, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing results in literature.

Numerical solution of Abel equation using operational matrix method with Chebyshev polynomials

2017 ◽  
Vol 10 (03) ◽  
pp. 1750053 ◽  
Author(s):  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Chebyshev Polynomials ◽  
Matrix Method ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Abel Equation ◽  
Operational Method ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Nonlinear Algebraic Equations

In this paper, we present a numerical scheme for solving the Abel equation. The approach is based on the shifted Chebyshev polynomials together with operational method. We reduce the problem to a set of nonlinear algebraic equations using operational matrix method. In addition, convergence analysis of the method is presented. Some numerical examples are given to demonstrate the validity and applicability of the method. The only a small number of Chebyshev polynomials is needed to obtain a satisfactory result.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

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