Asymptotic Solutions and Comparisons of a Generalized Van Del Pol Oscillator with Slowly Varying Parameter

A generalized Van del Pol oscillator with slowly varying parameter is studied. The leading order approximate solutions are obtained respectively by three methods and comparisons are made with numerical results. Different amplitudes are also made to compare the accuracy of the three methods.


INTRODUCTION
For general strongly nonlinear oscillator with slowly varying parameter, many perturbations are difficult to be applied strictly.The problem has caused many researcher's attention and been researched widely in recent years.This paper is to study the following strongly nonlinear oscillator of the form 0 ) , ( ) , ( is the slow scale.For some special cases of k and g , we can obtain Van del Pol oscillator, Rayleigh equation and pendulum equation.We assume that functions k and g are arbitrary nonlinear functions and Eq.( 1) has periodic solutions when 0 = ε .For the case of quadratic and cubic nonlinear function ) , ( t x g , Kuzmak-Luke multiple scales method [1][2][3][4] can be applied efficiently, and the asymptotic solutions expressed by Jacobian elliptic functions can also be obtained [4].For general nonlinear functions ) , ( t x g , Taylor series expansions are often used to approximate them but they are effective only for small amplitudes.Many efforts have been done to overcome the difficulty, such as Fourier series [3], equivalent linearization combined averaging method [5].Approximate potential method was first proposed by Li in Ref. [6] to deal with a generalized pendulum equation resulted from the free electron laser (FEL).In Ref. [6] the potential for the nonlinear oscillator is expressed by a polynomial of degree three such that the leading approximation is expressible in terms of elliptic functions.In Ref. [7] Cai first proposed equivalent nonlinearization method to overcome the difficulty of some kinds of nonlinearity.This method use quadratic or cubic nonlinear polynomial to approximate nonlinear function ) , ( t x g , and the least-squares fit method is used to decide the coefficients.Bosely presented a technique that uses numerical solutions to verify the order of the accuracy of an asymptotic expansion for several types of problems [8].
In this paper, we first obtain three approximate cubic nonlinear oscillators respectively by Taylor series expansions method, approximate potential method and equivalent nonlinearization method to approximate a generalized Van del Pol oscillator.Secondly, the leading order approximate solutions of these three approximate cubic nonlinear oscillators are obtained by the K-L multiple scales method.The numerical order verification is applied to verify that the asymptotic solutions are valid when the parameter ε is small for the three approximate cubic nonlinear oscillators but not uniformly valid for the original equation.The reason is that these three approximate cubic nonlinear oscillators have errors with the original equation.Finally, error analysis of the leading order approximate solutions shows that the errors are about one-tenth of the value of the small parameter ε .Error analysis also shows that Taylor series expansions method is better than approximate potential method and equivalent nonlinearization method when the amplitude is small, while equivalent nonlinearization method is better than Taylor series expansions method and approximate potential method when the amplitude is large.It also shows that Taylor series expansions method has large error for relatively large oscillations.

ASYMPTOTIC SOLUTION OF STRONGLY NONLINEAR OSCILLATOR
Van der Pol obtained the following equation 0 (2) which is negative damp for small oscillation, and positive damp for large oscillation.A modified Van del Pol oscillator has been recently proposed to describe a self-excited body sliding on a periodic potential [9]，which is described by the following equation Consider a generalized strongly nonlinear oscillator with slowly varying parameter in the form ) is the slow scale.Eq.( 4) can transform to 0 ) ( ) In order to obtain the asymptotic solution of Eq.( 5) by using K-L multiple scales method, we obtain three approximate cubic nonlinear oscillators respectively by Taylor series expansions method, approximate potential method and equivalent nonlinearization method.Next, we give a brief introduction to the three methods.

Approximate potential method
The fast scale + t , following Kuzmak [1], is defined as to be determined by the periodicity of the solution of Eq.( 5).Suppose that the solution of Eq. ( 5) can be developed in the multiple scales form 7) into ( 5) and equating powers of ε gives the leading order approximate equation 0 We obtain the energy integral where is the potential(For simplity, we set )and is the slowly varying energy of the system.According to the character of the potential function, we may fit a polynomial of degree four to the potential (10) such that the periodic solution is expressible in terms of elliptic functions, which will be discussed in detail later.

Equivalent nonlinearization method
According to the character of , we may seek a polynomial of the form The coefficients are chosen such that x and 2 x can be chosen around the center r x .
In the following, we choose as an example to consider.Following Taylor series expansions method, the approximate equation of Eq.( 5) becomes According to approximate potential method [6], the potential V is "U--Shaped", so Eq.( 11) has periodic solutions around 0 = r x . Denoting where the coefficients are chosen such that Substituting V for V in Eq.( 9), we can obtain the approximate equation of Eq.( 5) According to equivalent nonlinearization method [7], we consider the range of We now apply K-L multiple scales method to obtain the leading order approximate solutions of these three approximate cubic nonlinear oscillators.Firstly, we introduce the application of K-L multiple scales method in cubic nonlinear oscillators.For system (1), suppose that the solution can be developed in the form of asymptotic expression (7), where ) ( t ω will be determined by the periodicity of the solution of Eq.( 5).If the periodic is normalized to be 1, we have where r x is the resonance center.We denote where ( ) K v is the complete elliptic integral of the first kind associated with the modulus v and ) 1 ( is the complete elliptic integral of the second kind associated with the modulus v .More details of deduce, readers can refer to Ref. [10]. We assume the initial conditions are (0 11) and ( 23)、( 13) and ( 23)、( 14) and (23), using K-L multiple scales method, we obtain the following asymptotic solutions expressed by Jacobian elliptic functions Comparisons of the three leading order approximate solutions and numerical solutions of Eqs.( 5) and ( 23  So the accuracies of the three leading order approximate solutions are quite satisfactory.

NUMERICAL ORDER VERIFICATION OF APPROXIMATE SOLUTION
Following Bosley's technique of numerical order verification [8], we assume the solution of Eq.( The error of the asymptotic expansion is where K is a constant.Taking the logarithm of both sides of equation (28) yields In order to give a better overall estimation of difference between the exact (or numerical) and asymptotic solutions, instead of Ref. [8] with a fixed time 0 t t = , an average error is introduced in Ref. [11 are fixed points in the concerned domain of time t .To verify the order of asymptotic expansions(24)-( 26), we first find the numerical solutions of Eq.( 11), Eq.( 13), Eq.( 14) and ( 23 in Eq.( 28) is replaced by the numerical solutions of Eq.( 11), Eq.( 13) and Eq.( 14) respectively.Fig. 4-Fig.6 plot the values of the errors at these 34 points, and these points are nearly on a line.The leastsquare fit of the data is used to determine the slopes 1.00537, 1.00777, and 1.00736, which are in good agreement with the theoretical slope Therefore, we can conclude that K-L multiple scales method is valid when the parameter is small for the three approximate cubic nonlinear oscillators.If the exact solution ) , ( ε t x exact in Eq.( 28) is replaced by the numerical solution of Eq.( 5), the result is not asymptotic valid., respectively.And the result obtained by approximate potential method is not a line.The reason is that the three approximate cubic nonlinear oscillators have errors with Eq.( 5), and the errors bring larger error between the asymptotic solutions with the numerical solution of Eq.( 5).

ANALYSIS OF ERROR
Now we will show a numerical comparison of these three methods for parameter ε by using technique [8,11].Fig. 7   Note that the average error 0 F increases lineary as parameter ε increases, and the errors are about one-tenth of the small parameter ε .Obviously, approximate potential method has higher accuracy at this time.We can compare further the accuracy of the three methods for different amplitudes.a: Taylor expansions method b: approximate potential method c: equivalent nonlinearization method

CONCLUSIONS
K-L multiple scales method can be applied validly to strongly nonlinear oscillators with slowly varying parameter.For quadratic nonlinear oscillators, the asymptotic solutions expressed by Jacobian elliptic functions can be obtained.The error analysis also shows that the asymptotic solutions are still valid for 1 .0 = ε , which is not very small.
Taylor series expansions method is better when the amplitude is small, and equivalent nonlinearization method is better when the amplitude is large.It also shows that Taylor series expansions method has large error for large oscillations.

.
Therefore, when we graph ) log( N E versus ε log for different values of ε , these points should be nearly on a line and the linear equation that interpolates these points using a linear least-squares fit should have slope 1 + N .

Fig. 4 Fig. 6
Fig. 4 Numerical verification of the solution Fig. 5 Numerical verification of the solution (24) with ε starting from 0.001 to 0.1 (25) with ε starting from 0.001 to 0.1

.
For example, we consider ] The lines obtained by Taylor series expansions method and equivalent nonlinearization method are

Fig. 7 F and parameter ε ( 1
Fig. 7 The relation between the average error 0 F and parameter ε ( 1 ) 0 ( = x ) a: Taylor expansions method b: approximate potential method c: equivalent nonlinearization method

Fig. 8 F
Fig. 8 The relation between the average error Fig.9 The relation between the average error 0 F and parameter ε ( 5 .0 ) 0 ( = x )