scholarly journals Construction of Approximate Analytical Solutions to Strongly Nonlinear Coupled van der Pol Oscillators

2014 ◽  
Vol 6 ◽  
pp. 817570
Author(s):  
Y. H. Qian ◽  
W. K. Liu ◽  
S. M. Chen
Keyword(s):  
Analytical Solutions ◽  
Degrees Of Freedom ◽  
Van Der Pol Oscillator ◽  
Practical Significance ◽  
Engineering System ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Van Der Pol Oscillators

Using nonlinear theory to research vibration model of engineering system has important theoretical and practical significance. Multi-degree-of-freedom (MDOF) coupled van der Pol oscillator is a typical model in the nonlinear vibration; many complex dynamic problems in practical engineering can be simplified as this model to be solved in the end. This paper discusses a class of two-degrees-of-freedom (2-DOF) coupled van der Pol oscillator, which was divided into three parameters of different situations α1≠α2, β1≠β2, and γ1≠γ2 to discuss. Employing symbolic software such as Mathematica for those problems, the explicit analytical solutions of frequency ω and displacements x1( t) and x2( t) are well formulated. Results showed that the homotopy analysis method (HAM) can effectively deal with this kind of parameter of different coupled vibrators, just request the values of some parameters are not too big. Finally, we got four important theorems to simplify the solution of the nonlinear system.

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Approximate Solution ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Amplitude Spectrum ◽  
Van Der Pol Oscillator ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Series Expression

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

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.

HIGH-DIMENSIONAL CHAOS IN DISSIPATIVE AND DRIVEN DYNAMICAL SYSTEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2823-2869 ◽  
Author(s):  
Z. E. MUSIELAK ◽  
D. E. MUSIELAK
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Onset Of Chaos ◽  
General Rules ◽  
Van Der Pol Oscillators ◽  
Control And Synchronization

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

Analytical solutions of linear and nonlinear Klein-Fock-Gordon equation

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Sajida Rasheed
Keyword(s):  
Analytical Solutions ◽  
Gordon Equation ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Relativistic Version ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

AbstractIn this paper, we deal with some linear and nonlinear Klein-Fock-Gordon (KFG) equations, which is a relativistic version of the Schrödinger equation. The approximate analytical solutions are obtained by using the homotopy analysis method (HAM). The efficiency of the HAM is that it provides a practical way to control the convergence region of series solutions by introducing an auxiliary parameter }. Analytical results presented are in agreement with the existing results in open literature, which confirm the effectiveness of this method.

Perturbation Solution of an Oscillator With Periodic van der Pol Damping

1993 ◽  
Author(s):  
Duane W. Storti ◽  
Cornelius Nevrinceanu ◽  
Per G. Reinhall
Keyword(s):  
Periodic Solution ◽  
Limit Cycle ◽  
Variational Equation ◽  
Phase Locking ◽  
Perturbation Solution ◽  
Van Der Pol Oscillator ◽  
Transition Curve ◽  
Van Der Pol ◽  
Additional Stability ◽  
Van Der Pol Oscillators

Abstract We present a perturbation solution for a linear oscillator with a variable damping coefficient involving the limit cycle of the van der Pol equation (van der Pol 1926). This equation arises as the variational equation governing the stability of in-phase vibration in a pair of identical van der Pol oscillators with linear coupling. The van der Pol oscillator has served as the classic example of a limit cycle oscillator, and coupled limit cycle oscillators appear in mathematical models of self-excited systems ranging from tube rows in cross flow heat exchangers to arrays of stomates in plant leaves. As in many systems modeled by coupled oscillators, criteria for phase-locking or synchronization are of fundamental importance in understanding the dynamics. In this paper we study a simple but interesting problem consisting of a pair of identical van der Pol oscillators with linear diffusive coupling which corresponds, in the mechanical analogy, to a spring connecting the masses of the two oscillators. Intuition and earlier first-order analyses suggest that the spring will pull the two masses together causing stable in-phase locking. However, previous results of a relaxation limit study (Storti and Rand 1986) indicate that the in-phase mode is not always stable and suggest the existence of an additional stability boundary. To resolve the apparent discrepancy, we obtain a new periodic solution of the variational equation as a power series in ε, the small parameter in the sinusoidal van de Pol oscillator. This approach follows Andersen and Geer’s (1982) solution for the limit cycle of an isolated van der Pol oscillator. The coupling strength corresponding to the periodic solution of the variational equation defines an additional stability transition curve which has only been observed previously in the relaxation limit. We show that this transition curve, which provides a consistent connection between the sinusoidal and relaxation limits, is O(ε2) and could not have been delected in O(ε) analyses. We determine the analytical expression for this stability transition curve to O(ε31) and show very favorable agreement with numerical results we obtained using an Adams-Gear method.

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