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

2010 ◽  
Author(s):  
Youhua Qian
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Three 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.

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.

Periodic Solutions for Multi-Degree-of-Freedom Nonlinear Dynamical System Solved by the Extended Homotopy Analysis Method

2009 ◽  
Author(s):  
Youhua Qian ◽  
Wei Zhang ◽  
Shuping Chen
Keyword(s):  
Periodic Solutions ◽  
Numerical Integration ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical System ◽  
Time History ◽  
Nonlinear Problems ◽  
Nonlinear Dynamic System ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Homotopy Analysis

In normal circumstances, many practical engineering problems are nonlinear and can be described by multi-degree-of-freedom (MDOF) dynamical systems. Theoretically speaking, the exact solutions are very scarce, so it is extremely significant to develop the analytic tools for nonlinear systems in engineering. Inasmuch as the homotopy analysis method (HAM) can overcome the foregoing restrictions of conventional perturbation techniques, this method has been widely applied to solve a variety of nonlinear problems. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate periodic solutions for MDOF nonlinear dynamic system. The periodic solutions for the parametric excitation buckled thin plate system of MDOF are applied to illustrate the validity and great potential of this method. In addition, comparisons are conducted between the results obtained by the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the second-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

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