NONLINEAR OSCILLATORS WITH SLOWLY VARYING PARAMETERS

A generalized Krylov-Bogoliubov-Mitropolsky (KBM) method is extended for the study of strongly nonlinear oscillators with slowly varying parameters. The asymptotic amplitude and phase are derived and then the asymptotic solutions of arbitrary order are obtained theoretically. Cubic nonlinear oscillators with polynomial damping are studied in detail. Three examples are considered: a generalized Van der Pol oscillator, a Rayleigh equation and a pendulum with variable length. Comparisons are also made with numerical solutions to show the efficiency and accuracy of the present method. Keywordsstrongly nonlinear oscillation, KBM method, slowly varying parameter

is the slow scale.Assume that g and h are arbitrary nonlinear functions of their arguments and Eq.( 1) has periodic solutions when 0 = ε .Many problems in engineering are modeled as Eq.( 1), such as pendulum with varying length [1], machines with variable mass [2], and motion of electron in free-electron laser(FEL) [3].For the case of linear spring in ) , ( t y g , the classical KBM method [4] is effective to deal with Eq.(1).While ) , ( t y g is a cubic polynomial in y , the KB method is extended as the elliptic KB (EKB) method by using Jacobian elliptic functions [5,6], and adiabatic invariants coupled with elliptic functions are applied to find the asymptotic solutions [7].However, the EKB method can give only the first order approximation and the classical KBM method is effective only for weakly nonlinear oscillations.For the case of linear damping in ) , , ( t dt dy y h , Kuzmak proposed a multiple scales method to obtain the conditions of periodicity and asymptotic solutions of first order [8], and then Luke extended it to higher order [9].Kevorkian and Li reviewed and compared the Kuzmak-Luke method and that of near-identity averaging transformations [10].Bourland and Haberman used a two-variable procedure to give a careful analysis of Eq.( 1) and derived the equation governing the slowly varying phase [11], which has been summarized by Kevorkian and Cole [12].Recently, the author developed the Kuzmak-Luke method to obtain the asymptotic solutions of Eq.
(1) and applied it to quadratic and cubic nonlinear oscillators [13].For Eq. ( 1) without slowly varying parameters, that is a generalized KBM method is proposed to obtain the asymptotic solutions of arbitrary order [14].This method is effective for general nonlinear function ) ( y g . In this paper, this generalized KBM method will be developed to treat strongly nonlinear oscillators with slowly varying parameters.The asymptotic amplitude and phase are derived and then the asymptotic solutions of arbitrary order are obtained theoretically.As an application, cubic nonlinear oscillators with polynomial damping are studied in detail.
Three typical examples are considered: a generalized Van der Pol oscillator, a Rayleigh equation and a pendulum with variable length.Comparisons are also made with numerical solutions to show the efficiency and accuracy of the present method.

THE GENERALIZED KBM METHOD
In this section we extend the generalized KBM method [14] to Eq. (1).Assume that the asymptotic solutions of Eq.(1) can be expanded as where a is the amplitude, and ψ is the phase factor.
are periodic functions of ψ with a constant period normalized to be T .a and ψ satisfy For simplicity, the initial conditions are assumed to be From Eqs.( 3) and ( 4), we have the derivative transformations Substituting ( 2) into (1) and equating same powers of ε give the equations where n F are known functions, According to the generalized KBM method [14], we assume that Eq.( 8) has a periodic solution 0 y and 0 B can be found out.It is easy to verify that a solution of the homogeneous equation ( 9) has the form The other solution linearly independent of Ι y can be found by the reduction of order Using variation of parameters, the general solutions of Eq.( 9) are y and integrating from 0 to T with respect to ψ , we obtain the solvability conditions for Eq.( 9) Then n A and n B can be determined by Eqs.( 14) and ( 15) respectively.Particularly, A and 1 B can be worked out as and a and ψ can be solved from the following equations The procedure can be carried out up to desired order, although the calculations are rather involved.

APPLICATION TO CUBIC NONLINEAR OSCALLATORS
where n is a positive integer.For 2 ≥ n , the explicit approximations of Eq.( 21) are difficult to obtain by the multiple scales method [11,12,13].Suppose that the solution of Eq. ( 21) can be developed in the form of asymptotic expression (2).Note that ) , ( 0 Eq. ( 22) has an exact analytical solution expressed by Jacobian elliptic functions in the is the complete elliptic integral of the first kind with the modulus v .
Substituting Eq. ( 23) into Eq.( 22), we can find that By the formulas of elliptic integrals [15] , the solution of Eq. ( 22) can be expressed by Similarly, when 0 ) ( , the solution of Eq. ( 22) can be expressed by

CONCLUSIONS
The generalized KBM method is effective for strongly nonlinear oscillators with slowly varying parameters and can obtain asymptotic solutions of arbitrary order theoretically, while the classical KBM method works only for weakly nonlinear oscillations and the EKB method can give only the leading order approximation.
Cubic nonlinear oscillators are studied in detail to illustrate the present method.Three examples are considered: a generalized Van der Pol oscillator, a Rayleigh equation and a pendulum with slowly varying length.The asymptotic results are in good agreement with the numerical solutions.
elliptic integral of the second kind associated with the modulus v .Comparison of numerical solution and asymptotic amplitude obtained by Eqs.(19) and (25) with 01 .0 = ε is shown in Fig.1.In this paper the symbolic language Mathematica is used to implement the asymptotic and numerical solutions.