On the integrability of some forced nonlinear oscillators

Journal Pre-proof On the integrability of some forced nonlinear oscillators Maria V. Demina, Dmitry I. Sinelshchikov

PII: DOI: Reference:

S0020-7462(19)30927-8 https://doi.org/10.1016/j.ijnonlinmec.2020.103439 NLM 103439

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International Journal of Non-Linear Mechanics

Received date : 20 December 2019 Revised date : 3 February 2020 Accepted date : 3 February 2020 Please cite this article as: M.V. Demina and D.I. Sinelshchikov, On the integrability of some forced nonlinear oscillators, International Journal of Non-Linear Mechanics (2020), doi: https://doi.org/10.1016/j.ijnonlinmec.2020.103439. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

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On the integrability of some forced nonlinear oscillators a

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Maria V. Deminaa , Dmitry I. Sinelshchikova,∗ Department of Applied Mathematics, National Research University Higher School of Economics, Moscow, Russian Federation

Abstract

We consider integrability properties of a family of forced nonlinear oscillators, which generalizes the Li´enard equation. We demonstrate that some forced oscillators with previously known first integrals can be linearized via certain nonlocal transformations. Furthermore, we show that the whole family of Li´enard (n, n +1) equations with arbitrary external forcing admits a first integral. We study in detail the case of the Li´enard (3, 4) equation due to its value for applications. We prove that despite the fact that this equation possesses one Darboux first integral and can be linearized, it does not have an additional Darboux integral and, hence, is not Darboux integrable. Therefore, we demonstrate that certain nonlocal transformations do not preserve the property of Darboux integrability. Keywords: Forced oscillators; Duffing–Van-der-Pol oscillator; linearization; Darboux integrability.

1. Introduction

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In this article we consider the following family of non-autonomous forced nonlinear oscillators yzz + f (y)yz + g(y) + h(z) = 0, (1.1)

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where f 6≡ 0, g 6≡ 0 and h are arbitrary smooth functions. Integrability and dynamical properties of nonlinear oscillators have been attracting much attention in recent years [1–10]. Forced oscillators given by (1.1) are of great interest due to their applications in various field of science, such as physics, chemistry, and biology (see, e.g. [1, 11]). Lately, there has been some interest in finding equations of type (1.1) that posses certain first integrals (see, e.g. [12]). Here we show that the existence of first integrals obtained in work [12] can be explained by the linearizability of the corresponding equations via the generalized Sundman transformations. We also discuss integrability properties of some other forced oscillators, which also can be linearized via nonlocal transformations. We Corresponding author Email addresses: [email protected] (Maria V. Demina), [email protected] (Dmitry I. Sinelshchikov) Preprint submitted to International Journal of Non-Linear Mechanics

February 3, 2020

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show that despite this fact these oscillators are not Darboux integrable. Thus, we demonstrate that a certain class of nonlocal transformation does not preserve Darboux integrability. For other aspects of integrability of equations of type (1.1) and its generalizations see, e.g. [13–20] and references therein. The rest of this work is organized as follows. In the next section we provide linearizability conditions for family of equations (1.1) and discuss some of its particular cases. Section 3 deals with the Darboux integrability of a certain forced oscillator from family (1.1). In the last section we briefly summarize and discuss our results. 2. Linearizability conditions and first integrals of (1.1)

Linearizability conditions for equations of the form (1.1) to wζζ = 0

(2.1)

via the generalized Sundman transformations w = F (z, y),

dζ = G(z, y)dz,

(2.2)

can be obtained from the results of [21] (see, also [13, 22]), where a more general case was considered. Thus, one can show that (1.1) can be transformed into (2.1) if and only if f = C 1 gy +

1 , C1

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where C1 6= 0 is an arbitrary constant. At the same time functions G and F are defined by the following relations
C12 Ggy − C1 Gz − G Fy2 + C1 Fz GFyy = 0, GFyy − Fy Gy = 0,
C1 (h + g) Fy3 + C12 gy − 1 Fy2 Fz − C1 Fy2 Fzz + C1 Fyy Fz2 = 0,

(2.3)

(2.4)

C1 Fy2 gy − Fy Fzy + Fyy Fz = 0.

This system of equations can be considerably simplified. Consequently, from (2.4) we get     Z    z z z G = exp − Fy , Fz − C1 g + exp − h(z) exp dz Fy = 0. (2.5) C1 C1 C1

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In order to find linearizing transformations explicitly one needs to solve a linear first order partial differential equation for F , which can always be done locally under reasonable assumptions on functions h and g. Although it may be difficult to find an explicit global solution of this partial differential equation, one does not need to do it for constructing a first integral of (1.1). Indeed, equation (2.1) has a first integral wζ = C2 . From the expression for wζ via yz , y and z we get    Z    z z z exp C1 g + exp − h(z) exp dz + yz = C2 . (2.6) C1 C1 C1 Here C2 is an arbitrary constant. The above results can be summarized as follows 2

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Theorem 2.1. [13, 21, 22] If (2.3) holds, then the corresponding equation from family (1.1) possesses first integral (2.6) and can be linearized to (2.1) via (2.2).

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Let us consider an example of a linearizable forced oscillator. Suppose that f = 3y 2 + α. Then from (2.3) at C1 = 1/ we find that g = y 3 + (α − )y. This case corresponds to the forced Duffing–Van der Pol oscillator yzz + (3y 2 + α)yz + y 3 + (α − )y + h(z) = 0,

whose first integral can be easily obtained from (2.6). The result is  Z     z 3 I = y + (α − )y + dz exp{−z} + yz exp{z}. h(z) exp C1

(2.7)

(2.8)

This first integral was found in [12], where it was shown that the forced Duffing–Van der Pol oscillator did not possess another independent Darboux first integral. a

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Figure 1: Contour plots of first integral (2.13) for different forcing functions and values of the parameters: (a) h(z) = sin(5z), δ = 4.05, β = −1, C2 = 0.001; (b) - h(z) = sin(z), δ = 4, β = −1, C2 = 1; (c) h(z) = cos(5z), δ = 4.5, β = −1, C2 = 0.001.

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Now we consider more examples of forced oscillators with a first integral. Suppose that f (y) = α(n + 1)y n , where n ∈ N and α 6= 0 is an arbitrary parameter. Then from (2.3) at C1 = 1 we find that g = αy n+1 −y+β, where β is an arbitrary parameter. The corresponding first integral can be found from (2.6) and has the form    Z    z z z n+1 exp C1 αy − y + β + exp − h exp dz + yz = C2 . (2.9) C1 C1 C1 Consequently, the following family of Li´enard equations with arbitrary external force yzz + α(n + 1)y n yz + αy n+1 − y + β + h(z) = 0

possesses first integral (2.9). Therefore, the following statement holds 3

(2.10)

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where β is an arbitrary constant.

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Corollary 2.1. Forced Li´enard equations with the polynomial coefficients of degrees (n, n+1) and arbitrary forcing function admit a first integral (2.10) provided that Z g = f dy − y + β, (2.11) Within the family of Li´enard (n, n + 1) equations there are several equations which are important for applications. One of these cases is Li´enard (3, 4) equation, which has the form yzz + (y 3 + βy + δ)yz + y(y 3 + 2βy + 4δ − 16) + h(z) = 0.

(2.12)

Here β and δ are arbitrary parameters. Equation (2.12) has drown some attention from both theoretical and applied points of view. For example, a Liouvillian first integral in the autonomous case was derived in article [8]. As far as other applications are concerned, equation (2.12) can be considered as a traveling wave reduction (z = x − δt) of the following reaction-diffusion-convection equation ut = uxx +(u3 +βu)ux +u(u3 +2βu+4δ −16)+h(x−δt) or as a stationary (z = x) reduction of the following reaction-diffusion-convection equation ut = uxx + (u3 + βu + δ)ux + u(u3 + 2βu + 4δ − 16) + h(x). See also [23] for other applications of (2.12). Further our aim is to study the case of Li´enard (3, 4) equation in detail. b

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a

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Figure 2: Contour plots of first integral (2.13) for different forcing functions and values of the parameters: (a) h(z) = exp{−5z}z, δ = 4, β = −1, C2 = 1; (b) h(z) = exp{−z 2 }z, δ = 4.1, β = −10, C2 = 1; (c) h(z) = z, δ = 10, β = −10, C2 = −1.

One can see that the coefficients of equation (2.12) satisfy condition (2.3) and, thus, it has the first integral  Z   y 3 I = exp{4z} yz + (y + 2βy + 4δ − 16) + h(z) exp{4z}dz exp{−4z} . (2.13) 4 Let us consider some properties of first integral (2.13). We demonstrate the plots of (2.13) for different periodic forcing functions and values of parameters in Fig. 1. One can see that there are several possible types of solutions of (2.12) including periodic solutions. We also 4

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3. Darboux integrability

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demonstrate the plots of (2.13) for different non-periodic forcing functions and values of parameters in Fig.2. It shows that in the case of non-periodic forcing periodic solutions are less frequent, though are also possible. In the next section we consider the Darboux integrability of forced nonlinear oscillator (2.12) in details. We show that despite the fact that (2.12) can be linearized via (2.2) it is not Darboux integrable.

The aim of the present section is to study the existence of an additional independent first integral for nonlinear oscillators of the previous section. We shall deal with the generalized Darboux first integrals. Let D be a simply connected open subset of C. By M(D) we denote the field of meromorphic functions in D. In addition, we shall consider an algebraic extension Ma (D) of the field M(D) and the rings of polynomials in the variables u and v with coefficients in M(D) or Ma (D). These rings will be denoted as M(D)[u, v] and Ma (D)[u, v]. In what follows we suppose that h(z) is a meromorphic in D function and f (y) and g(y) are polynomials over the field C. Only those extensions of the field M(D) that contain elements meromorphic in some subset of full Lebesgue measure in D are interesting from practical point of view. The following operator X =

∂ ∂ ∂ + yz − {f (y)yz + g(y) + h(z)} ∂z ∂y ∂yz

(3.1)

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defines the derivative with respect to equation (1.1). Let F (u, v, z) be a polynomial in Ma (D)[u, v] \ Ma (D). The zero set F (y, yz , z) = 0 is an invariant surface of equation (1.1), if X F = λ(y, yz , z)F , where λ(y, yz , z) ∈ Ma (D)[y, yz ]. An invariant surface is supposed to be irreducible if the corresponding polynomial is irreducible in Ma (D). A function of the form I(y, yz , z) = f1d1 . . . fKdK exp[W (y, yz , z)],

d1 , . . . , dK ∈ C,

K ∈ N ∪ {0},

(3.2)

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where f1 (y, yz , z), . . ., fK (y, yz , z) ∈ Ma (D)[y, yz ] and W (y, yz , z) ∈ Ma (D)(y, yz ), is called the generalized Darboux function [24, 25]. It can be proved that a generalized Darboux function given by (3.2) is a first integral of ordinary differential equation (1.1) if and only if f1 (y, yz , z) = 0, . . ., fK (y, yz , z) = 0 are invariant surfaces of the equation in question [24, 25]. Let us define a functional Puiseux series in a neighborhood of the point y = ∞ as the series of the form +∞ X l0 −l (3.3) w(y, z) = bl (z)y n0 , l=0

where l0 ∈ Z, n0 ∈ N, and the coefficients {bl (z)} belong to an algebraic extension of a field Ma (u1 , . . . , uM ) with u1 (z), . . ., uM (z) being from the field Ma (D). 5

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Suppose that U (y, yz , z) is a formal sum of the monomials c(z)y q1 yzq2 , where q1 , q2 ∈ Q and q1 +q2 ≤ L for some L ∈ N. Let us introduce the operator of projection {U (y, yz , z)}+ giving the sum of the monomials of U (y, yz , z) with integer non–negative powers: q1 , q2 ∈ N ∪ {0}. In other words, {U (y, yz , z)}+ yields the polynomial part (with respect to y and yz ) of U (y, yz , z). The following theorem was proved in article [12]. Theorem 3.1. Let F (y, yz , z) = 0 with F (y, yz , z) ∈ Ma (D)[y, yz ] \ M(D), Fyz 6≡ 0 be an irreducible invariant surfaces of equation (1.1). Then F (y, yz , z) takes the form ( ) N Y F (y, yz , z) = µ(y, z) {yz − wj (y, z)} , N ∈ N, (3.4) j=1

+

where µ(y, z) ∈ Ma (D)[y] and w1 (y, z), . . ., wN (y, z) are pairwise distinct functional Puiseux series in a neighborhood of the point y = ∞ that solve the partial differential equation wz + wwy + f (y)w + g(y) + h(z) = 0.

(3.5)

Moreover, the degree of F (y, yz , z) with respect to yz does not exceed the number of distinct functional Puiseux series of the from (3.3) satisfying equation (3.5) whenever this number is finite.

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It follows from this theorem that in order to find all the irreducible invariant surfaces of equation (1.1) one needs to construct all the functional Puiseux series near the point y = ∞ that satisfy equation (3.5) and to consider all possible combinations of these series as given in (3.4). In what follows we shall in fact work with formal functional Puiseux series. The operators of differentiation in the field of formal functional Puiseux series are defined as formal operators with most of the properties similar to those valid for (uniform) convergent functional Puiseux series. As soon as an irreducible invariant surface is found the (uniform) convergence of the corresponding functional Puiseux series follows from the implicit function theorem. The method of Puiseux series which we use in this article was introduced in articles [9, 12, 26]. The main result of this section is the following theorem.

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Theorem 3.2. If the function h(z) and the field Ma (D) are such that the following condition Z h(z)e4z dz ∈ Ma (D) is valid, then the unique independent generalized Darboux first integral

of nonlinear oscillator (2.12) is given by (2.13). As a result this oscillator is not integrable with two independent generalized Darboux first integrals. Proof. We begin the proof of the theorem by finding all the irreducible invariant surfaces of nonlinear oscillator (2.12). Substituting F = F (y, z) into the partial differential equation Fz + yz Fy − {(y 3 + βy + δ)yz + (y 3 + 2βy + 4δ − 16)y + h(z)}Fyz = λ(y, yz , z)F

(3.6)

and equating the coefficients at yz yields the equation Fy = λ1 (y, z)F , where the cofactor should be represented as λ(y, yz , z) = λ1 (y, z)yz + λ0 (y, z). The unique family of polynomial 6

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in y solutions of this equation does not depend on y. Consequently, we conclude that there are no invariant surfaces that do not depend on yz . Now suppose that F (y, yz , z) = 0 with Fyz 6≡ 0 is an irreducible invariant surface of nonlinear oscillator (2.12). Balancing the highest–order terms (with respect to yz ) in equation (3.6), we find the similar equation µy = λ1 (y, z)µ, were µ(y, z) is the highest–order coefficient of the polynomial F (x, y, z) and the representation of the cofactor is the same. Analogously, we can state that the cofactor λ does not depend on yz and the polynomial µ given in (3.4) does not depend on y. Without loss of generality, we can suppose that the polynomial µ neither depends on z. Equation (3.5) takes the form wz + wwy + (y 3 + βy + δ)w + (y 3 + 2βy + 4δ − 16)y + h(z) = 0.

(3.7)

Analyzing this equation, we see that there exist exactly two dominant balances that produce functional Puiseux series in a neighborhood of the point y = ∞. These balances and solutions of related equations take the following form (I) :

w(wy + y 3 ) = 0,

(II) :

y 3 (w + y) = 0,

y4 ; 4 w(y, z) = −y. w(y, z) = −

(3.8)

Let us find the Fuchs indices related to these balances and to the corresponding series. For this aim we need to calculate the Gˆateaux derivatives E0 [b0 (z)y r + sB(z)y r−j , y; z] − E0 [b0 (z)y r , y; z] δE0 [b0 (t)y r ] = lim s→0 δw s = L(z; j)B(z)y r˜,

(3.9)

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where (I): E0 [w, y, z] = w(wy + y 3 ), (II): E0 [w, y, z] = y 3 (w + y) and the function w(y, z) = b0 (z)y r solves the equation E0 [w, y, z] = 0, see (3.8). In expression (3.9) L(z; j) is a linear differential operator. Recall that if there exists j0 ∈ Q such that L(z; j0 ) ≡ 0, then j0 is called a Fuchs index. For more details on calculating Fuchs indices and functional series for partial differential equations (see, e.g. [27, 28]). In the case (I) the corresponding functional Puiseux series has the unique Fuchs index j0 = 4. The compatibility condition for the series to exist is automatically satisfied and we get the family of functional generalized Puiseux series with an arbitrary coefficient at y 0 . The balance of type (II) does not have Fuchs indices. Consequently, all its coefficients are uniquely determined. Substituting functional Puiseux series with undetermined coefficients and the leading–order behavior given by (3.8) into equation (3.7), we conclude that these series are in fact functional Laurent series and take the form (I) :

(II) :

∞

X y 4 βy 2 + (4 − δ)y + bl (z)y 3−l ; w(y, z) = − − 4 2 l=3 w(y, z) = −y +

∞ X

al (z)y

l=2

7

1−l

.

(3.10)

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It follows from theorem 3.1 that the polynomial F (y, yz , z) can be factorized in the following way ) (N −k   Y y 4 βy 2 (j) + − (4 − δ)y − b0 (z) − . . . (yz + y − . . .)k . (3.11) F (y, yz , z) = yz + 4 2 j=1 +

We have introduced the parameter k in this expression in order to indicate that the series of type (II) either appears in the factorization of invariant surfaces or does not. Using irreducibility of the polynomial F (y, yz , z), we conclude that the series w1 (y, z), . . ., wN −k (j) (y, z) are pairwise distinct. Consequently, the coefficients b0 (z), j = 1, . . . , N − k should be pairwise distinct and k = 0 or k = 1. Indeed, the invariant surface possessing two or more coinciding functional Puiseux series in factorization (3.11) is not irreducible. We start with the case k = 1. Calculating several first coefficients of the functional Puiseux series under consideration, we require that the non–polynomial part of the expression in brackets in (3.11) vanishes. This gives necessary conditions for invariant surfaces to exist. Finding the coefficients at yzN −1 y −l , l ∈ N in the representation of F (y, yz , z) yields the system of equations N −1 X (j) al+1 (t) + bl+3 (t) = 0, l ∈ N. (3.12) j=1

This system is composed of algebraic equations and ordinary differential equations. For the (1) (N −1) sake of convenience, we introduce the functions Bm (t) = {b0 (t)}m + . . . + {b0 (t)}m , where m ∈ N. It is straightforward to derive the following relations (j) (j) b0 b0, z ;

B2, zz = 2

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B2, z = 2

N −1 X

N −1  X

(j) (j) b0 b0, zz

+

j=1

j=1

(j) {b0, z }2



;

...

(3.13)

We take the first thirteen equations of system (3.12) and see that they are inconsistent. Consequently, there are no invariant surfaces with k = 1. Now let us consider the case k = 0. Introducing the new variable u according to the rule

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y 4 βy 2 u = yz + + − (4 − δ)y, 4 2

(3.14)

we can rewrite the polynomial F˜ (y, u, z) = F (y, yz , z) giving invariant surfaces in the following form (N )  Y (j) F˜ (y, u, z) = u − b0 (z) . (3.15) j=1

(j) b0 (z)

+

If all the functions are from the field Ma (D), then irreducible invariant surfaces of (j) equation (2.12) are represented by the polynomials F˜ (y, u, z) = u − b0 (z) with u given by (3.14). These invariant surfaces exist provided that the series of type (I) terminates at 8

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Integrating this equation yields

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zero term. The latter happens whenever the function b0 (z) satisfies the following ordinary differential equation b0,z + 4b0 + h(z) = 0 (3.16)

b0 (z) = − exp(−4z)

Z

exp(4z)h(z)dz

(3.17)

According to the condition of the theorem the function b0 (z) belongs to the field Ma (D). Consequently, the zero–set of the polynomial Z y 4 βy 2 (3.18) + − (4 − δ)y + exp(−4z) exp(4z)h(z)dz F (y, yz , z) = yz + 4 2 gives the unique family of irreducible invariant surfaces of equation (2.12). The cofactor we find by direct substitution of the expression (3.18) into equation (3.6). This yields λ(y, yz , z) = −4. Any invariant surface with the cofactor being a function of z produces a first integral of the original equation. In our case this first integral is a generalized Darboux function given by (2.13). Now let us suppose that there exists another generalized Darboux first integral H(y, yz , z) independent with I(y, yz , z). We have proven that the family of invariant surfaces represented by polynomial (3.18) is unique. Consequently, the new integral should take the form Z z K  Y y 4 βy 2 + − (4 − δ)y + exp(−4z) exp(4s)h(s)ds H(y, yz , z) = yz + 4 2 z 0 k=1 αk

× exp[R(y, yz , z)],

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(3.19)

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where {α1 . . . αK }, {C1 . . . CK } are from the field C and R(y, yz , z) ∈ Ma (D)(y, yz ). Expressing the variable yz from the integral I(y, yz , z) and substituting the result into H(y, yz , z), we can without loss of generality suppose that an independent with I(y, yz , z) generalized ˜ yz , z) = exp[R(y, ˜ yz , z)], where R(y, ˜ yz , z) ∈ Darboux first integral takes the form H(y, ˜ yz , z) is a rational first inteMa (D)(y, yz ). Further, it is straightforward to verify that R(y, gral of equation (2.12). In this case the numerator and the denominator of the first integral ˜ yz , z) should be invariant surfaces of the original equation. But according to the preR(y, vious results there exists only one family of invariant surfaces. Hence, the first integral ˜ yz , z) is not independent with I(y, yz , z). It is a contradiction. R(y, This completes the proof.

4. Conclusion

In this work we have considered a family of nonlinear forced oscillators. We have presented a subfamily of this family that can be linearized via nonlocal transformations and, 9

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5. Acknowledgments

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consequently, admits a first integral. In particular, we have shown that the Li´enard (n, n+1) equations with arbitrary forcing function possess a first integral provided certain correlations hold. Furthermore, we have considered the case of the Li´enard (3, 4) equation in details and have proved that this equation is not Darboux integrable although it admits one nonautonomous Darboux first integral. Therefore, we have demonstrated that certain class of nonlocal transformations does not preserved Darboux integrability.

This word was supported by Russian Science Foundation grant 19-71-10003. References

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A family of forced nonlinear oscillators admitting certain first integral is found A first integral for the Liénard (n,n+1) equation with arbitrary forcing is constructed Darboux non-integrabiliy of the Liénard (3,4) equation is proved

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Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: