Analytical approximations for a conservative nonlinear singular oscillator in plasma physics

A modified variational approach and the coupled homotopy perturbation method with variational formulation are exerted to obtain periodic solutions of a conservative nonlinear singular oscillator in plasma physics. The frequency–amplitude relations for the oscillator which the restoring force is inversely proportional to the dependent variable are achieved analytically. The approximate frequency obtained using the coupled method is more accurate than the modified variational approach and ones obtained using other approximate methods and the discrepancy between the approximate frequency using this coupled method and the exact one is lower than 0.31% for the whole range of values of oscillation amplitude. The coupled method provides a very good accuracy and is a promising technique to a lot of practical engineering and physical problems.     

On the limit cycles,period-doubling,and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator

In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.     

Analytical approximate solutions for the generalized nonlinear oscillator

He's energy balance method (HEBM) is employed in this article to obtain the analytical approximate solution of the generalized nonlinear oscillator. Existence of periodic solutions is analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A number of numerical simulations are carried out and accuracy of the HEBM is then examined within an error analysis. The exact values of the natural frequency numerically obtained via the elliptic integrals are taken into account as the references bases and the relative error is then evaluated for a range of oscillation amplitudes. Excellent correlation of the approximate frequencies with the exact ones demonstrates that the approximate solutions are quite consistent even for large amplitudes of oscillation.     

Construction of approximate periodic solutions to a modified van der Pol oscillator

A new iteration scheme is proposed and applied for the modified van der Pol oscillator. A simple and effective iteration procedure to search for the periodic solutions of the equation is given. This procedure is a powerful tool for the determination of the approximate frequencies and periodic solutions of the nonlinear differential equations. The solutions obtained using the present iteration procedure are in good agreement with the numerical integration obtained by a fourth order Runge–Kutta method, which shows the applicability of the procedure.     

Accurate analytical solutions to nonlinear oscillators by means of the Hamiltonian approach

The purpose of this paper is to apply the Hamiltonian approach to nonlinear oscillators. The Hamiltonian approach is applied to derive highly accurate analytical expressions for periodic solutions or for approximate formulas of frequency. A conservative oscillator always admits a Hamiltonian invariant, H , which stays unchanged during oscillation. This property is used to obtain approximate frequency–amplitude relationship of a nonlinear oscillator with high accuracy. A trial solution is selected with unknown parameters. Next, the Ritz–He method is used to obtain the unknown parameters. This will yield the approximate analytical solution of the nonlinear ordinary differential equations. In contrast with the traditional methods, the proposed method does not require any small parameter in the equation. Copyright © 2011 John Wiley & Sons, Ltd.     

Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturbation method

In this paper He’s homotopy perturbation method has been adapted to calculate higher-order approximate periodic solutions for a nonlinear oscillator with discontinuity for which the elastic force term is an anti-symmetric and quadratic term. We find that He’s homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Just one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 0.73% for all values of oscillation amplitude, while this relative error is as low as 0.040% when the second iteration is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance method reveals that the former is very effective and convenient.     

On algorithms for estimating computable error bounds for approximate periodic solutions of an autonomous delay differential equation

Machine tool chatter has been characterized as isolated periodic solutions or limit cycles of delay differential equations. Determining the amplitude and frequency of the limit cycle is sometimes crucial to understanding and controlling the stability of machining operations. In Gilsinn [Gilsinn DE. Computable error bounds for approximate periodic solutions of autonomous delay differential equations, Nonlinear Dyn 2007;50:73–92] a result was proven that says that, given an approximate periodic solution and frequency of an autonomous delay differential equation that satisfies a certain non-criticality condition, there is an exact periodic solution and frequency in a computable neighborhood of the approximate solution and frequency. The proof required the estimation of a number of parameters and the verification of three inequalities. In this paper the details of the algorithms will be given for estimating the parameters required to verify the inequalities and to compute the final approximation errors. An application will be given to a Van der Pol oscillator with delay in the non-linear terms.     

On the Solution of High Dimensional Fokker Planck Equations using Orthogonal Polynomial Expansion

Solving the Fokker-Planck-Equation for multidimensional nonlinear systems is a great challenge in the field of stochastic dynamics. As for many mechanical systems a general idea about the shape of stationary solutions for the probability density function is known, it seems promising to use an approach that contains this knowledge. This is done using a Galerkin-method which applies approximate solutions as weighting functions for the expansion of orthogonal polynomials, e.g. generalized Hermite polynomials [1]. As examples, nonlinear oscillators containing cubical restoring (Duffing oscillators) and cubical damping elements are considered. The method is applied to the two-dimensional problem of a single-degree-of-freedom oscillator and consecutively extended up to dimension ten. Results for probability density functions are presented and compared with results from Monte Carlo simulations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.     

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.     

Active nonlinear saturation-based control for suppressing the free vibration of a self-excited plant

A nonlinear saturation-based control strategy for the suppression of the free vibration of a self-excited plant is presented. The self-excitation to have the classical form of that of the van der Pol oscillator is considered. The control technique is implemented by coupling the active absorber with the plant via a specific set of quadratic nonlinearities. The perturbation method of multiple scales is employed to find the first-order approximate solutions to the governing equations. Then a stability analysis is conducted for the response of the system and the performance of the control strategy is investigated. A parametric investigation is carried out to see the effects of changing the damping ratio of the absorber, and the value of the feedback gain on the responses of the plant and the absorber. Finally, the perturbation solutions are verified by numerical integration of the governing differential equations. It is demonstrated that the saturation-based control method is effective in reducing the vibration response of the self-excited plant when the absorber’s frequency is exactly tuned to one-half the natural frequency of the plant.     

Some properties of approximate solutions for vector optimization problem with set-valued functions

In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions and weak efficient solutions are discussed. In particular, we prove the connectedness of the set of approximate solutions under the condition that the objective functions are quasiconvex set-valued functions.     

Reduced-order model for laminar vortex-induced vibration of a rigid circular cylinder with an internal nonlinear absorber

The nonlinear interaction of a laminar flow and a sprung rigid circular cylinder results in vortex-induced vibration (VIV) of the cylinder. Passive suppression of the VIV by attaching an internal nonlinear vibration absorber that acts, in essence, as a nonlinear energy sink (NES) to the cylinder has been observed in finite-element computations involving thousands of degrees of freedom (DOF). A single-DOF self-excited oscillator is developed to approximate the limit-cycle oscillation (LCO) of the cylinder undergoing VIV. This self-excited oscillator models the interaction of the flow and the cylinder. Then, a two-DOF reduced-order model for the system with the internal NES is constructed by coupling the single-DOF NES to the single-DOF self-excited oscillator. Hence, the complicated high-dimensional system of flow-cylinder-NES involving thousands of DOF is reduced to a two-DOF model. The two targeted energy transfer mechanisms responsible for passive VIV suppression that are observed in the finite-element computations are fully reproduced using the two-DOF reduced-order model. This reduction of the dynamics to an easily tractable low-dimensional reduced-order model facilitates the approximate analysis of the underlying dynamics. Moreover, the underlying assumptions of the order reduction, and the parameter ranges of validity of the reduced-order model are formulated and systematically studied.     

Bifurcation analysis of a simple 3D oscillator and chaos synchronization of its coupled systems

Tama?evi?ius et al. proposed a simple 3D chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated.     

Decomposition method for the Camassa–Holm equation

The Adomian decomposition method is applied to the Camassa–Holm equation. Approximate solutions are obtained for three smooth initial values. These solutions are weak solutions with some peaks. We plot those approximate solutions and find that they are very similar to the peaked soliton solutions. Also, one single and two anti-peakon approximate solutions are presented. Compared with the existing method, our procedure just works with the polynomial and algebraic computations for the CH equation.     

Iterative reproducing kernel method for nonlinear oscillator with discontinuity

In this paper, iterative reproducing kernel method is applied to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. The solution obtained by using the method takes the form of a convergent series with easily computable components. An illustrative example is given to demonstrate the effectiveness of the present method. The results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the nonlinear oscillator with discontinuities.     

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

Approximate limit cycles of coupled nonlinear oscillators with fractional derivatives

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) coupled nonlinear oscillators with fractional derivatives. Approximate limit cycles (LCs) of two systems of the coupled fractional van der Pol (VDP) oscillators and the fractional damped Duffing resonator driven by a fractional VDP oscillator are exampled for illustrating the validity and great potential of the HAM. The presented approach can provide approximate LCs very accurately and efficiently compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. Furthermore, it is capable of tracking unstable LCs which cannot be generated by some time-marching numerical algorithm. Based on the obtained results, we analyze effect of different fractional orders, coupling coefficient, and nonlinear coefficient of the coupled equations on amplitudes and frequencies of the LCs.     

On Approximate Solutions in Vector Optimization Problems Via Scalarization

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze’s approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems. AMS Classification:90C29, 49M37 This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.