Analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol-Duffing oscillators by extended homotopy analysis method

2011 ◽  
Vol 219 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
Y. H. Qian ◽  
W. Zhang ◽  
B. W. Lin ◽  
S. K. Lai
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

scholarly journals Application of Extended Homotopy Analysis Method to the Two-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. H. Qian ◽  
S. M. Chen ◽  
L. Shen
Keyword(s):  
Homotopy Analysis Method ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Different Types ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems

2018 ◽  
Vol 28 (04) ◽  
pp. 1850049 ◽  
Author(s):  
H. X. Fu ◽  
Y. H. Qian
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

Modified Homotopy Analysis Method for Nonlinear Aeroelastic Behavior of Two Degree-of-Freedom Airfoils

2016 ◽  
Vol 16 (09) ◽  
pp. 1520001 ◽  
Author(s):  
Yaobin Niu ◽  
Zhongwei Wang ◽  
Dequan Wang ◽  
Bing Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Control Parameter ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Aerodynamic Load ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Two Degree Of Freedom ◽  
Convergence Control Parameter

In this paper, the homotopy analysis method (HAM) is extended to deal with the nonlinear aeroelastic problem of a two degree-of-freedom (DOF) airfoil. To avoid determination of the parameter for the complicated high-order minimization problem, a new modified HAM is proposed based on the idea of minimizing the squared residual. Using this method, the convergence-control parameter is determined by the low order squared residual of the governing equations, and then the problem is solved in a way similar to the basic HAM. The proposed method is used to solve the nonlinear aeroelastic behavior of a supersonic airfoil, with the unsteady aerodynamic load evaluated by the piston theory. Two examples are prepared, for which the frequencies and amplitudes of the limit cycles are obtained. The approximate solutions obtained are demonstrated to agree excellently the numerical solutions, meanwhile, the convergence-control parameter can be easily determined using the present approach.

Periodic Solutions for Coupled Van Der Pol Oscillators of Two-Degree-of-Freedom Solved by Homotopy Analysis Method

2009 ◽  
Author(s):  
Wei Zhang ◽  
Youhua Qian ◽  
Qian Wang
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Two Degree Of Freedom

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter ħ. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

scholarly journals THE HOMOTOPY ANALYSIS METHOD IN BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1230024 ◽  
Author(s):  
ANDREA BEL ◽  
WALTER REARTES
Keyword(s):  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential ◽  
Bifurcation And Stability

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

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