Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation

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1 Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation D.K.K. Adjaï a, L. H. Koudahoun a, J. Akande a, Y.J.F. Komahou b and M. D. Monsia a 1 a Deartment of Physics, University of Abomey- Calavi, Abomey-Calavi, 01.B.P.56, Cotonou, BENIN monsiadelhin@yahoo.fr b Deartment of Industrial and Technical Sciences, ENSET-Lokossa, University of Abomey, Abomey, BENIN komaf@yahoo.fr This aer shows that exlicit and exact general eriodic solutions for various tyes of Liénard equations can be comuted by alying the generalized Sundman transformation. As an illustration of the efficiency of the roosed theory, the cubic Duffing equation and Painlevé- Gambier equations were considered. As a major result, it has been found, for the first time, that equation XII of the Painlevé-Gambier classification can exhibit, according to an aroriate arametric choice, trigonometric solutions, but with a shift factor. keywords: Liénard equations, cubic Duffing equation, Painlevé-Gambier equations, Jacobian ellitic functions, exact eriodic solution, generalized Sundman transformation. Introduction A large art of hysics, engineering and alied science roblems is solved by using linear models where the rincile of suerosition may be alied. However, this consideration is only valid as a first aroximation, since a more realistic aroach in the mathematical modeling of many rocesses and henomena in hysics and alied science leads to consider nonlinear roblems often exressed in terms of nonlinear differential equations. The henomena of oscillation,dielectric olarization, cree deformation and stress relaxation under loading in dynamic structures are examles of roblems where the nonlinearity roerties occur in different asects [1]. In structural dynamics, for examle, geometric nonlinearities are often due to large dislacements or large deformations inducing not only stiffening but also nonlinear otical roerties of the structure. The material nonlinearity is relative to the stressstrain constitutive law which imlies the henomena of dissiation of energy in heat, and the inertial nonlinearity results often from the quadratic terms in the first order derivative in the equations of motion or deformation [1, ]. Thus, the failure to take into account these nonlinearities can ermanently affect the erformance of these material structures during their service, which can become a danger for users and the environment. In this sense, good design of these structures or reair after damage requires to calculate the exlicit and exact general solutions of the nonlinear differential equations describing their dynamics. In this ersective the exlicit and exact general solutions of these nonlinear differential equations are more aroriate from the hysical oint of view as well as engineering alication viewoint than aroximate and numerical solutions. In so doing, the eriodic solutions constitute a articular class of solutions since many henomena in the real world, such as mechanical oscillations of conservative systems, are of a eriodic nature. Thus a secial interest must be accorded to the exlicit and exact general eriodic solutions. However, it should be emhasized that there is no master mathematical method that works in all cases of calculation of exlicit and exact general eriodic solutions. It suffices to note the large and rich variety of mathematical methods existing in the literature for this urose. Therefore the fundamental roblem of finding exlicit and exact general eriodic solutions to nonlinear differential equations constitutes yet 1 corresonding author. E.mail: monsiadelhin@yahoo.fr 1

2 an active research field of mathematics. It is also known that such solutions are only ossible for a few number of nonlinear differential equations. There aears then logic to be interested to mathematical theories which may rovide with simlicity and elegance exlicit and exact general eriodic solutions to nonlinear differential equations. In the rocess of solving nonlinear second order differential equations, it is convenient to consider the linearization methods which transform the initial equation into a linear differential equation whose roerties are well known. As such, one can note methods like oint transformation, contact transformation and generalized Sundman transformation. The latter, according to Euler and Euler [3], is a more general transformation than the first two transformations of linearization. The generalized Sundman transformation is a nonlocal transformation that has been frequently used to determine the first integrals and solutions of a variety of Liénard-tye differential equations [3 5]. Recently, Akande et al. [6] have formulated a generalized Sundman transformation alied to the harmonic oscillator equation to demonstrate for the first time, the existence of a class of quadratic Liénard tye equations whose solutions are trigonometric functions, but with amlitude deendent frequency. Thus, it has been shown that equation XVIII of the Painlevé-Gambier classification [7] and its inverted version and other quadratic Liénard tye equations admit trigonometric eriodic solutions by alying the generalized Sundman transformation develoed. It is also well known that the study of oscillation in many mechanical systems leads to consider the cubic Duffing equation consisting to introduce the cubic term in the dislacement into the harmonic oscillator equation of the linear structural model in order to take into account the geometric nonlinearity roerties. Recently it is found that the theoretical investigation of viscoelastic deformation in material structures taking into consideration simultaneously the geometric, material and inertial nonlinearities, leads to nonlinear roblems described in terms of mixed or quadratic Liénard tye equations [8]. One may note in this context that the quadratic Liénard tye differential equations and the cubic Duffing equation have imortant alications in hysics. The cubic Duffing equation lays an interesting role in many areas of hysics like classical and quantum mechanics [9]. In this regard the forced cubic Duffing equation is often, for examle, used to illustrate a number of nonlinear system characteristics such as jums, subharmonic and suerharmonic resonances in addition to the amlitude deendent frequency exhibited already by the free cubic Duffing oscillator. Let us now consider the cubic Duffing equation and equations XIX and XII of the Painlevé-Gambier classification. The general solution of the unforced cubic Duffing equation has often been calculated by means of secific methods based on the fact that this equation is an ellitic equation, to say, the Jacobian ellitic functions are solutions of this equation. On the other hand, Painlevé-Gambier equations XIX and XII have recently been studied by the alication of generalized Sundman transformation [4]. The authors [4] were able to calculate the first integrals in addition to those already determined by Ince [7], and arametric solutions for these equations. However, no exlicit and exact general eriodic solution has been comuted for these equations by the generalized Sundman transformation used by those authors [4]. The cubic Duffing equation belongs to the usual class of Liénard equations, whereas the Painlevé-Gambier XII and XIX equations are of the class of the quadratic Liénard tye equations. According to the classification given by Ince [7], the equation XII is a canonical equation of tye II, whereas the equation XIX is a canonical equation of tye III. Although these equations have been already studied in the literature with more or less of success, it is imortant, to further identify their analytical roerties, to investigate them by different tyes of analytical methods, with also a view on the simlicity, elegance and extent of alication of the methods concerned. In other words, it is reasonable to ask whether other integration methods could not be alied for such equations, since these new methods may aear easy and more elegant to be alied in the resolution of the roblem and may lead to new solutions in some cases. The roblem now is to show that the exlicit and exact general eriodic solutions can be comuted for these nonlinear differential equations by alying the generalized Sundman transformation roosed by Akande et al. [6]. The fundamental question to be solved, in the context of the generalized Sundman transformation roosed by Akande et al. [6], can then be formulated as follows: Can we calculate the exlicit and exact general eriodic solutions of the equations considered above? The resent work suoses this fulfillment by alying the generalized Sundman transformation roosed by Akande et al. [6]. This fact will allow testing the efficiency of the roosed generalized Sundman transformation to ensure the calculation of exlicit and exact general eriodic solutions to a wide variety of nonlinear ordinary differential equations that are often encountered in mathematical modeling of hysics and alied science roblems. The resent theory will have the advantage of generalizing the aroach develoed by Akande et al. [6] and other methods that exist in literature in the ersective of a large class of nonlinear differential equations of the Liénard tye for which exlicit and exact general eriodic solutions can be comuted by a well-documented linearization method, the generalized Sundman transformation. As ractical imlications, these results may be useful in numerical exeriments involving these equations. To comute the exlicit and exact general eriodic solutions of these equations, a brief review of the theory introduced by Akande and his coworkers [6] is given section ) and secondly, the generalized theory under question is carried out section 3). Finally the exlicit and exact general eriodic solutions to unforced cubic Duffing equation and for some Painlevé-Gambier equations are comuted section 4) and, a discussion and conclusion for the research contribution is carried out.

3 1 Review of the theory by Akande and coworkers [6] Let us consider the general second order linear differential equation as y τ) + by τ) + a yτ) = 0 1.1) where rime denotes differentiation with resect to τ, a and b are arbitrary arameters. By alication of the generalized Sundman transformation with F t, x) yτ) = F t, x), dτ = Gt, x)dt, Gt, x) 0 1.) x F t, x) = gx) l dx, Gt, x) = exγϕx)) where l and γ are arbitrary arameters, and gx) 0 and ϕx) are arbitrary functions of x, to 1.1), one may obtain ẍ + l g x) gx) γϕ x))ẋ + bẋ exγϕx)) + a exγϕx)) gx) l dx = 0 1.3) gx) l as general class of mixed Liénard tye differential equations. The arametric choice b = 0, leads to ẍ + l g x) gx) γϕ x))ẋ + a exγϕx)) gx) l dx = 0 1.4) gx) l as general class of quadratic Liénard tye nonlinear equations. An interesting case of 1.4) is obtained by considering l = 0, viz ẍ γϕ x)ẋ + a x exγϕx)) = 0 1.5) The imortance of 1.5) is that it exhibits trigonometric functions as exact eriodic solutions but with amlitudedeendent frequency. In [6] it is shown for the first time that some existing Liénard tye equations may exhibit exact trigonometric eriodic solutions. One may also find that the quadratic Liénard tye equation ẋ ẍ x + x 1 + x = 0 1.6) used to model oscillation of a liquid column in a U-tube [10] belongs to the class of equations defined by 1.5) by utting ϕx) = 1 ln1+x), and γ = 1, so that this equation may exhibit trigonometric functions as exact eriodic solution [11]. That being so the general theory under question may be formulated. General theory The objective of this section is to extend the receding theory to look for large classes of linearizable Liénard tye equations in order to ensure exact solutions which may be exressed exlicitly or by quadratures. In the revious theory the general second order linear equation is considered in the form of homogeneous equation with no forcing function. Instead in this section, let us consider the general second order linear equation 1.1) with a constant forcing term, y τ) + by τ) + a yτ) = c.1) where c is an arbitrary arameter. The alication of the generalized Sundman transformation 1.) to.1) yields ẍ + l g x) gx) γϕ x))ẋ + bẋ exγϕx)) + a exγϕx)) gx) l dx c exγϕx)) = 0.) gx) l gx) l The comarison of 1.3) with.) shows that the constant forcing function c contributes to the general class of mixed Liénard tye equation 1.3) by an additional term c exγϕx)). This establishes the extension of the gx) l theory by Akande et al. [6] to a wider class of Liénard tye nonlinear differential equations. The question now is to ask: What haens to equations 1.4) and 1.5) with resect to this additional term? Let us consider then b = 0 and l = 0. The arametric choice b = 0 leads to ẍ + l g x) gx) γϕ x))ẋ + a exγϕx)) gx) l dx c exγϕx)) = 0.3) gx) l gx) l 3

4 The comarison of 1.4) with.3) shows also the ersistence of the receding additional term in the generalized class of quadratic Liénard tye equations reresented by.3). The alication of l = 0 to.3) gives the class of quadratic Liénard tye equations ẍ γϕ x)ẋ + a x exγϕx)) c exγϕx)) = 0.4) which differs from the class of quadratic Liénard tye equations 1.5) by the additional term c exγϕx)). By noting yτ) = A 0sinaτ + α) + c a.5) where a 0, the general solution to.1) with b = 0, the general solution to.4) becomes where A 0 and α are arbitrary arameters, and τ = φt) satisfies xt) = A 0sinaφt) + α) + c a.6) dt = ex γϕx))dφt).7) The solution.6) remains of eriodic form. For a convenient choice of ϕx) and γ the solution.6) may exhibit harmonic eriodic behavior but with a shift factor c. This extends therefore the theory by Akande et al. [6]. It a is then convenient to show the ability of the current theory to rovide exlicit and exact general eriodic solutions to some existing Liénard equations. 3 Alications of general theory This section is devoted to show the usefulness of the resent theory by considering some well known nonlinear equations. 3.1 Cubic Duffing equation The cubic Duffing equation is one of the most investigated equations from mathematical oint of view as well as hysical viewoint. The motivation results from the fact that this equation arises in mathematical modeling of many roblems of mechanics and quantum mechanics, for examle. The cubic Duffing equation is of the form [1] where ω 0 and are arbitrary arameters. ẍ + ω 0x + x 3 = 0 3.1) From the mechanical viewoint, the arameter > 0 is called hardening arameter while < 0 is called softening arameter, and = 0, gives the well known linear harmonic oscillator equation. The equation 3.1) may be recovered from the general class of quadratic Liénard equations.3). Indeed, the functional choice ϕx) = lnfx)) leads to ẍ + l g x) gx) γ f x) fx) )ẋ + a fx) γ gx) l dx cfx)γ = 0 3.) gx) l gx) l which for fx) = x and gx) = x, yields The arametric choice l = γ = 1, reduces 3.3) to which is the Duffing equation 3.1) by noting c = ω0, and a 3.1) may be written according to.5) ẍ + l γ) ẋ x + a l + 1 x4γ+1 cx 4γ l = 0 3.3) xt) = ε ẍ cx + a x3 = 0 3.4) ω 0 =. So the general solution to the Duffing equation + sinaφt) + α) 3.5) where dφt) εdt = ω0 + A 0 sinaφt) + α) 3.6) 4

5 and ε = ±1. The evaluation of the integral resulting from 3.6) dφt) J = ω0 + A 0 sinaφt) + α) deends on the value of ω 0 and. Therefore three distinct cases are considered in this aer [13]. 3.7) Case 1: > 0, and ω 0 < 0 In this case where > 0 and ω0 < 0, it is moreover assumed that ω 0 > A 0 > 0. By noting [13] = ω0 + A 0 and the integral J becomes [13] = δ = arcsin A 0 A 0 ω 0 1 sinaφ + α) 3.8) 3.9) J = 1 F δ, ) 3.10) a where F δ, ) is the ellitic integral of the first kind so that using 3.6) sin δ = sn aε ) A 0 t + C), where snz, k) designates a Jacobian ellitic function and C an arbitrary arameter. In this way to say 1 sinaφ + α) = sn aε ) A 0 t + C), sinaφ + α) = 1 sn aε ) A 0 t + C), from which the general solution to the Duffing equation 3.1) becomes which reduces to 3.11) 3.1) [ xt) = ε ω 0 + 4sn aε 1 t + C), )] 3.13) xt) = ε A 0 [ 1 sn aε 1 A 0 t + C), )] 3.14) Using the identity k sn z, k) + dn z, k) = 1, the relation 3.14) becomes immediately xt) = ε ) A 0 aε dn t + C), 3.15) Knowing that a = ε, 3.15) may be written in the form xt) = ε ) A 0 dn t + C), 3.16) By making A = A 0, and Ω = A 0, Ω = A, the general solution to 3.1) definitively reads xt) = εadn Ωt + C), ) 3.17) where dnz, k) is a Jacobian ellitic function, and = A +ω0 ). The solution 3.17) differs from that obtained A by some revious authors [14] with a secific method. This is in concordance with the fact that for an ellitic equation like the Duffing equation, different ellitic functions may be found as solutions [15]. 5

6 Case : < 0, and ω 0 > 0 For < 0 and ω0 > 0, it is also assumed that ω 0 > A 0 > 0. So, the general solution to the Duffing equation 3.1) takes, using 3.17), the form xt) = εadc Ωt + C), ) ) where A = A 0, and Ω = A. The function dcz, k) is a Jacobian ellitic function. This result differs from that comuted by some revious authors [14]. Case 3: > 0, and ω 0 > 0 In this case, it is assumed that 0 < ω 0 < A 0, 0 < ω 0 < A 0. So, the integral J may be written as [13] J = F δ, 1 aa 0 ) 3.19) where δ = arcsin [ ] A 01 sinaφ + α)) A 0 ω0 3.0) has the receding value, and F δ, 1 ) denotes the ellitic integral of the first kind. In this regard the following relationshi may be written, taking into consideration 3.6) [ A 01 sinaφ + α)) = sn aε A 0t + C), 1 ] A 0 ω0 3.1) which may be rewritten sinaφ + α) = 1 [ ω 0 sn aε A 0t + C), 1 ] A 0 sinaφ + α) = 1 sn aε A 0t + C), 1 ) 3.) 3.3) In this ersective the general solution to 3.1) reads xt) = ε ω sn aε A 0t + C), 1 ) which may take the exression xt) = ε A 0 cn a A 0t + C), 1 ) 3.4) Knowing a = ε, A = A 0, and Ω = a A 0, Ω = A, or Ω = A + ω0, the general solution to Duffing equation 3.1) reduces immediately to the form xt) = εacn Ωt + C), 1 ) 3.5) where cnz, k) denotes a Jacobian ellitic function. It can be noted that this result is in agreement with that found in [14] by a secific method. It would be now interesting to consider in the sequel of this work some Painlevé-Gambier equations. 3. Painlevé-Gambier XII equation This subsection is intended to carry out the general solution to the Painlevé-Gambier XII equation [7] ẍ ẋ x qx3 x r δ x = 0 3.6) 6

7 for r = δ = 0, to say ẍ ẋ x qx3 x = 0 3.7) The reduced Painlevé-Gambier equation 3.7) may be found from.4) by choosing ϕx) = ln x, γ = 1, a = q, and c =. Therefore the general solution to 3.7) takes the exression where ε = ±1, and dt = dφt) xt) xt) = A 0 sinε qφt) + α) q dφt) dt = A 0 sinε qφt) + α) q 3.8) 3.9) By integration the quantity J = dφt) A 0 sinε qφt) + α) q leads to consider two distinct cases following the value of qa ) Case 1: q A 0 < 1 In this case the integral J becomes [13] such that q J = q ε 3 A 0 q ε qφt) + α = tg 1 tg 1 1 q A 0 ε tg qφt)+α) tg ε ) qa 0 1 q A 0 q 3 A 0 qt + C) q 3.31) + q 3.3) So the general solution to 3.7) becomes xt) = A 0 sin tg 1 1 q A 0 tg ε q 3 A 0 qt + C) q + q q 3.33) The solution xt) is real for q < 0. In this way, the solution 3.33) may clearly exhibit a harmonic eriodic behavior with a shift factor q. Case : q A 0 > 1 This case corresonds to [13] J = q ln ε q q3 A 0 which gives according to 3.9) ε qφt) + α = tg 1 q A 0 1 ε tg qφt)+α) ) qa 0 q A 0 tg ε qφt)+α) ) qa 0 + q A ex ε q 1 ex ε q q q3 A 0 q q3 A 0 t + C))) ) ) t + C)) + q 3.35) Therefore the general solution xt) may be exressed in the form q A ex ε xt) = A 0 sin tg 1 q 1 ex ε q q q3 A 0 q q3 A 0 )) t + C) ) + q q t + C)) 3.36) 7

8 For q < 0, the solution 3.36) is real and then may exhibit also a harmonic eriodic behavior with a shift factor q. Consider now as final illustrative examle a generalized Painlevé-Gambier XIX equation. 3.3 Generalized Painlevé-Gambier XIX equation The Painlevé-Gambier XIX equation is of the form [7] A general form of 3.37) may be written as ẍ 1 ẋ x 4x x = ) ẍ 1 ẋ x + a x cx = ) so that for a = 4, and c =, one may recover the Painlevé-Gambier XIX equation. The equation 3.38) may be obtained from.4) by setting γ = 1, and ϕx)=ln 4 x. Thus the solution to 3.38) immediately takes the exression following.6) xt) = A 0 sinaφt) + α) + c 3.39) a where dφt) dt = 3.40) A 0 sinaφt) + α) + c a This equation is of the same form as 3.6). So three distinct cases may be distinguished. Case 1: a > 0, c > 0, and c a > A 0 > 0 In this case [13] and so the integral becomes = δ = arcsin J = a A 0 a A 0 + c ) 1 sinaφ + α) 3.41) 3.4) dφt) 3.43) A 0 sinaφt) + α) + c a J = a A 0 F δ, ) 3.44) where F δ, ) designates the Jacobian ellitic integral of the first kind. So according to 3.40) one may write a A 0 t + C) = F δ, ) sin δ = sn a ) A 0 t + C), In this regard, the general solution 3.39) becomes ) xt) = a dn t + C), 3.45) 3.46) which may take the exression where A = A 0, Ω = a A 0 xt) = A dn [Ωt + C), ] 3.47) Ω = aa, and may be rewritten as = a A c) a A 8

9 Case : a < 0, c > 0 and 0 < c a < A 0 In such a situation, the integral [13] where δ = arcsin A 0 F δ, 1 ) 3.48) J = 1 a ) a A 0 1 sinaφ+α)) a a A 0, and = A 0 +c a A 0. So the use of 3.40) leads to +c a t + C) = F δ, 1 ) 3.49) which may give from which a A 01 sinaφ + α)) = sn a a A 0 + c t + C), 1 ) sinaφ + α) = 1 a A 0 + c sn a a A 0 t + C), 1 ) 3.50) 3.51) Therefore the general solution 3.39) may take the exression xt) = [1 sn The general solution 3.5) may be also exressed as a t + C), 1 ) xt) = cn a t + C), 1 )] 3.5) 3.53) which becomes where A = A 0. xt) = A cn aa t + C), 1 ) For a = i a, a = ±i a, the general solution xt) may take the form which becomes definitively where Ω = xt) = A cn xt) = i ) a A t + C), 1 A 3.54) [ ] 3.55) cn a A t + C), 1 1 A xt) = cn [Ωt + C), 1 1 ] 3.56) a A, and i is the urely imaginary number. By making a = 4, and c =, the exact doubly eriodic solution to Painlevé-Gambier XIX equation may be written as where A = A 0 1, and = 1 A A + 1 A xt) = [ A cn + 1t + C), A xt) = cn [Ωt + C), A +1 A +1 A +1 A +1 ] 3.57) ] 3.58) 9

10 Case 3: a > 0, c < 0 and 0 < c a < A 0 This case corresonds also to = a A 0, δ = arcsin a A 0 +c J = 1 a ) a A 0 1 sinaφ+α)) a A 0, and the integral +c A 0 F δ, 1 ) 3.59) So the general solution 3.39) may be written as xt) = cn a t + C), 1 ) 3.60) xt) = A cn Ωt + C), 1 ) 3.61) where and A = 3.6) Ω = aa 3.63) The arameter may also be exressed as = a A c) a A. That being so it is then ossible to show the equivalence between the Duffing equation and the generalized Painlevé-Gambier XIX equation. 4 Equivalence between equations This section is devoted to highlight the mathematical equivalence between the Duffing equation and the generalized Painlevé-Gambier XIX equation. The comarison of 3.6) with 3.40) as well as the comarison of resulting general solutions suggest this mathematical equivalence, to say the maing of the Duffing equation onto the generalized Painlevé-Gambier XIX equation and vice versa, the maing of the generalized Painlevé-Gambier XIX equation into the Duffing equation. In other words, the general solution to the Duffing equation may be obtained in terms of the solution to the generalized Painlevé-Gambier XIX equation and vice versa. Formally consider the variable transformation x = W 4.1) due to the above general solutions to Duffing equation and general solutions to the generalized Painlevé-Gambier equation 3.38). The substitution of 4.1) into 3.1) yields Ẅ 1 Ẇ W + 4W + ω0w = 0 4.) which is the generalized Painlevé-Gambier XIX equation 3.38) by taking a = 4, and c = ω 0. So with that the equivalence between Duffing equation and the generalized Painlevé-Gambier XIX equation has been shown and a discussion may be formulated for the resent work. 5 Discussion In the theory of nonlinear differential equations, the roblem of calculating exlicit and exact general eriodic solutions often arises acutely, given the non-existence of a master solving method. In this work, it was roosed to check the efficiency of a generalized Sundman transformation suosed to solve certain tyes of nonlinear second order differential equations. Thus, the well known cubic Duffing equation in various branches of hysics such as classical and quantum mechanics has been chosen to illustrate this urose given, to our knowledge, that a generalized Sundman transformation has never been used to calculate its exlicit and exact general eriodic solutions. Usually the exlicit general solution of the cubic Duffing equation is comuted through secific methods which are based on the fact that all the twelve Jacobian ellitic functions are solutions of this equation [15]. By alication of the roosed linearization transformation, exlicit and exact general eriodic solutions of the cubic Duffing equation were comuted, as exected, as Jacobian ellitic functions with simlicity and elegance in 10

11 the solution stes, which characterize the generalized Sundman transformation. In this ersective, exlicit and exact general eriodic solutions of Painlevé-Gambier XIX equation have been comuted for the first time using a generalized Sundman transformation, as Jacobian ellitic functions, so that it has been ossible to show the mathematical equivalence between the Painlevé-Gambier XIX equation and the cubic Duffing equation. On the other hand, it was ossible to calculate exlicit and exact general trigonometric solutions but with a shift factor to Painlevé-Gambier XII equation for the first time in accordance with a suitable arametric choice. With these convincing results, it would be ossible now to conclude this research contribution. Conclusion The vital roblem of finding exlicit and exact general eriodic solutions to nonlinear differential equations is still an active research field of mathematics. Exlicit and exact general eriodic solutions are comuted for various tyes of Liénard equation by means of a generalized Sundman linearization aroach. In this way, the exlicit and exact general eriodic solutions to the cubic Duffing equation as well as for some Painlevé-Gambier equations are with simlicity and elegance determined. In so doing, it has been demonstrated for the first time that for suitable arametric choice equation XII of the Painlevé-Gambier classification exhibits trigonometric eriodic solutions but with a shift factor, which is a major finding in the investigation of the analytical roerties of the Painlevé-Gambier equations. It is also for the first time that exlicit and exact general eriodic solutions have been calculated for Painlevé-Gambier XIX equation by a generalized Sundman transformation as Jacobian ellitic functions. In this ersective as an interesting result, it has been shown that the cubic Duffing equation is mathematically equivalent to the generalized Painlevé-Gambier XIX equation such that the solution of the last equation may be obtained from the solution of the former and vice versa. Thus it has been shown that the generalized Sundman transformation can be a owerful mathematical tool in the determination of exlicit and exact general solutions to the roblems of hysics and alied science which are described in terms of nonlinear differential equations. References [1] Y.J.F. Komahou, M.D. Monsia, Hardening nonlinearity effects on forced vibration of viscoelastic dynamical systems to external ste erturbation field, Int.J. Adv. Al. Math. and Mech. 3) 015) [] G. Kerschen, K. Worden, A.F. Vakakis, J-.C. Golinval, Past, Present and future of nonlinear system identification in structural dynamics, Mechanical Systems and Signal Processing, Vol. 0, N 0. 3, , 006. [3] Norbert EULER and Marianna EULER, Sundman symmetries of nonlinear second-order and third-order ordinary differential equations, Journal of Nonlinear Mathematical Physics, vol.11, N ), [4] Partha Guha, Brun Khanra, A. Ghose Choudoury, On generalized Sundman transformation method, first integrals, symmetries and solutions of equations of Painlevé-Gambier tye, Nonlinear Analysis 7 010) [5] W. Nakim and S. V. Meleshko, Linearization of second-order ordinary differential equations by generalized Sundman transformations, symmetry, integrability and geometry: Methods and Alications, SIGMA 6 010), 051. [6] J. Akande, D.K.K. Adjaï, L.H. Koudahoun, Y.J.F. Komahou, M.D. Monsia, Theory of exact trigonometric eriodic solutions to quadratic Liénard tye equations, vixra: V3 017). [7] E.L. Ince, Ordinary differential equations, Dover, New York [8] M.D. Monsia, A mathematical model for redicting the relaxation of cree strains in materials, Physical Review and Research International, 3)01), [9] W. P. Reinhardt and P.L. Walker,Chater : Jacobian ellitic functions,in NIST Handbook of Mathematical Function, Editor, Franck W.J. Olver, 010. [10] W.R.Utz, Periodic solutions of a nonlinear second order differential equation, SIAM J. Al. Math, 191), [11] J. Akande, D.K.K. Adjaï, L.H. Koudahoun, Biswanath Rath, Pravanjan Mallick, Rati Ranjan Sahoo, Y.J.F. Komahou, Marc D. Monsia, Exact quantum mechanics of quadratic Liénard tye oscillator equations with bound states energy sectrum, vixra: V1 017). [1] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations John Wiley and Sons New York), 1979). [13] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Seventh edition,

12 [14] M. Lakshmanan, S. Rajaseekar, Nonlinear Dynamics, Integrability, Chaos and Patterns, Sringer-Verlag, Berlin, 1st edition, 003. [15] W. Schwalm, Ellitic functions sn, cn, dn, as trigonometry. P resentation.../handout.df 1