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On the complete integrability and linearization of nonlinear ordinary differential equations - Part V: Linearization of coupled second order equations

Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled SNODEs. Our procedure gives several new types of linearizing transformations of both invertible and non-invertible kinds. In both the cases we provide algorithms to derive the general solution of the given SNODE. We illustrate the theory with potentially important examples.


Introduction
Continuing our study on the integrability and linearization of coupled second order nonlinear ordinary differential equations (SNODEs), in the present paper, we focus our attention on the linearization of two coupled SNODEs. The present study arises not only for the completeness of Part IV (Chandrasekar et al. 2009) but also to show the importance of unfinished tasks that exist in the theory of linearization of two coupled SNODEs. As far as the first point is concerned, we show that one can also solve a class of coupled second order nonlinear ODEs by transforming them into two second order free particle equations and from the solutions of the latter one can construct the solution of the former, even though this is a nontrivial problem in many situations (one can also transform coupled nonlinear ODEs into uncoupled nonlinear ones, which has already been pointed out by us in the previous paper, Part IV). Regarding the second point, we wish to stress the fact that linearization of coupled nonlinear ODEs is a vast area of research which is still in its early stage. In this paper, we show that in spite of the difficulties that exist in this topic, one can make useful progress on certain issues, namely (i) developing a method to deduce all linearizing transformations wherein the new dependent variables are functions of the old dependent and independent variables only and not derivatives of the dependent variables, and (ii) developing a method of constructing solutions of nonlinear ODEs from the linear ones in the case of non-point transformations.
Even though the modern theory of linearization of nonlinear ODEs had originated and developed with the works of Lie, Tresse and Cartan (Mahomed & Leach 1989;Steeb 1993;Olver 1995;Ibragimov 1999;Chandrasekar et al. 2005), the entire subject was lying dormant for more than a century. Only recently, during the past two decades or so, notable progress has been made to linearize nonlinear ODEs through non-point (Duarte et al. 1994) or generalized transformations (Chandrasekar et al. 2006). For example, focussing our attention on single second order ODEs, generalized Sundmann (Euler et al. 2003;Euler et al. 2004) and generalized linearizing transformations (Chandrasekar et al. 2006) have been introduced to linearize a class of equations that cannot be linearized by invertible point transformations. As far as two coupled SNODEs are concerned, to our knowledge, most of the studies were focussed only on invertible point transformations irrespective of whether it is an analytical approach or a geometrical formulation. For a survey on this topic one may refer the recent papers of Merker (2006) and Mahomed & Qadir (2007) and for the earlier works in this direction we cite Crampin et al. (1996), Fels (1995), Grossman (2000), Soh & Mahomed (2001) and Qadir (2007). More recently Sookmee & Meleshko (2008) proposed a new algorithm to linearize the coupled second order ODEs by sequentially reducing the order of the equation.
In this work we aim to give a new dimension to the theoretical development of linearization of nonlinear dynamical systems having two degrees of freedom by proving that one can unearth a wide class of linearizing transformations besides invertible point transformations. Of course, the latter ones form a sub-class of the new ones which we construct in this paper. In the present study, we not only derive several new types of linearizing transformations but also propose systematic procedures to derive the general solution in all these cases. We also wish to emphasize here that we derive all these transformations from the first two integrals alone and thereby establish a potentially simple, straightforward and powerful approach in the theory of differential equations.
The plan of the paper is as follows. In §2 a we briefly describe the method of deriving linearizing transformations for a system of two coupled second-order ODEs. We show that one can have two classes of linearizing transformations, depending upon the nature of the independent variables. If the new independent variables are the same (z 1 = z 2 ), we put them in Class-A category, and if they are different (z 1 = z 2 ), then we put them in Class-B category. In §2 b(i) we consider Class-A category and identify three types of linearizing transformations. In §2 b(ii) we consider Class-B category and identify six types of linearizing transformations. In §3 we consider one specific example for each of the nine types of linearizing transformations we have identified, and obtain general solution to each one of them to demonstrate our procedure. Finally, we present our conclusions in §4.

Article submitted to Royal Society
Any transformation of the form T {t , x} defined by x, y,ẋ,ẏ)dt, which transforms the given set of nonlinear ODEs (2.1) to the free particle equations, is called a linearizing transformation in the present work. Let x, y,ẋ,ẏ), and I 2 = G(t, x, y,ẋ,ẏ).
(2.4) be the two first integrals of motion of the coupled system (2.1) and that they can be explicitly found, if they exist, for example, by using generalized modified Prelle-Singer (PS) method formulated in Paper IV (Chandrasekar et al. 2009). Then the following theorem ensures that the transformation can be deduced from the I i 's, i = 1, 2.
Theorem: Suppose a given nonlinear system R{t , x} of ODEs (2.1) is linearizable to a system of two uncoupled free particle equations through the linearizing transformation T {t , x} of the form (2.2), then the latter can be deduced from the first integrals I i (t, x, y,ẋ,ẏ), i = 1, 2.
Proof: Let us re-express each of the functions F and G in (2.4) as a product of two new functions, that is, x, y), x, y).
(2.5) Again rewriting f 3 and f 4 as total time derivatives of another set of functions, say z 1 and z 2 , respectively, that is dz 1 /dt = f 3 (t, x, y,ẋ,ẏ) and dz 2 /dt = f 4 (t, x, y,ẋ,ẏ), equation (2.5) can be further recast as Now identifying the functions f 1 (t, x, y) = w 1 and f 2 (t, x, y) = w 2 as the new dependent variables, equation (2.6) can be further recast in the form It may be noted that in general the new dependent variables, w 1 and w 2 , may also involveẋ andẏ, that is w 1 = G 1 (t, x, y,ẋ,ẏ) and w 2 = G 3 (t, x, y,ẋ,ẏ), and this possibility may lead us to identify more generalized transformations like pointcontact and generalized-contact transformations. However, in this paper we will confine ourselves only to the forms of w 1 and w 2 given by (2.2).

(b) The nature of transformations
An important question which we will focus upon in the present paper is that what are the possible forms of linearizing transformations one can unearth through the above procedure. We recall here that in the case of scalar SNODEs, one has point, generalized Sundman and generalized linearizing transformations (Chandrasekar et al. 2005;2006). As far as the coupled SNODEs (2.1) are concerned, since there are two independent variables z 1 and z 2 as given in equation (2.2), one can choose them to be either the same, z 1 = z 2 (Class-A), or different, z 1 = z 2 (Class-B). In the case of Class-A transformations, one can construct three different types of linearizing transformations, while for class-B one can formulate six different types of linearizing transformations, as we point out below. However, we also note that even further types of local transformations involving the variablesẋ anḋ y are possible but these are not included in the present study.
1. Suppose z 1 = z 2 = z is a perfect differential function and w i 's, i = 1, 2, and z do not contain the variablesẋ andẏ, then we call the resultant transformation, namely, w 1 = f 1 (t, x, y), w 2 = f 2 (t, x, y), z = f 3 (t, x, y), as a point transformation of type-I.
2. On the other hand, if z is not a perfect differential function, and w i 's, i = 1, 2, and z do not contain the variablesẋ andẏ, then we call the resultant transformation, namely, w 1 = f 1 (t, x, y), w 2 = f 2 (t, x, y), z = f 3 (t, x, y)dt, as a generalized Sundman transformation of type-I.
3. As a more general case, if we consider the independent variable z to contain the derivative terms also, that is x, y,ẋ,ẏ)dt, then we call the resultant transformation as a generalized linearizing transformation of type-I.
In our analysis, we do not consider the possibility w 1 = f 1 (t, x, y), w 2 = f 2 (t, x, y) and z = f 3 (t, x, y,ẋ,ẏ) because the procedure to handle it is different from the presently discussed linearizing transformations. This possibility will be studied separately.
1. If z 1 and z 2 are perfect differential functions and w i 's and z i 's, i = 1, 2, do not contain the variablesẋ andẏ, then we call the resultant transformation, namely, x, y), as a point transformation of type-II.
2. Suppose z 1 is a perfect differential function and z 2 is not a perfect differential function or vice versa, and if z 1 and z 2 do not contain the variablesẋ anḋ y, then we can call the resultant transformation, namely x, y), as a mixed point-generalized Sundman transformation.
3. On the other hand, if any one of the independent variables contains the variablesẋ andẏ, then we call the resultant transformation, namely x, y,ẋ,ẏ)dt and z 2 = f 4 (t, x, y), as a mixed point-generalized linearizing transformation.
4. Suppose the independent variables are not perfect differential functions and are also not functions ofẋ andẏ, that is, x, y)dt, then we call the resultant transformation as a generalized Sundman transformation of type-II.
5. Further, if one of the independent variables, say z 1 , does not contain the derivative terms while the other independent variable z 2 does contain the derivative terms or vice versa, that is x, y,ẋ,ẏ)dt, z 2 = f 4 (t, x, y)dt, then we call the resultant transformation as a mixed generalized Sundman-generalized linearizing transformation.
6. As a general case, if we allow both the independent variables, z 1 and z 2 , to be non-perfect differential functions and also to contain derivative terms, that is, x, y,ẋ,ẏ)dt, then the resultant transformation will be termed as a generalized linearizing transformation of type-II.
Finally the possibility that x, y,ẋ,ẏ) and z 2 = f 4 (t, x, y,ẋ,ẏ) is not considered in the present study and will be pursued separately.

Applications
In this section, we consider specific examples and illustrate each one of the linearizing transformations identified in the previous section so as to make clear the applicability of them under different situations.
(a) Class-A Linearizing Transformations (z 1 = z 2 ) (i) Example:1 Point transformation of Type-I Let us consider the system of SNODEs, where ω is a arbitrary constant. The two first integrals associated with (3.1), which can be obtained using the formulation given in §2 of Part IV (Chandrasekar et al. 2009), can be written as Rewriting (3.2) in the form (2.5)-(2.6), we get so that we can identify point transformation of type I as Utilizing the transformation (3.4), one can transform (3.1) to a set of free particle equations, namely d 2 w1 dz 2 = 0 and d 2 w2 dz 2 = 0, so that w 1 = I 1 z + I 3 and w 2 = I 2 z + I 4 , where I 3 and I 4 are integration constants. Substituting the expressions for w i 's, i = 1, 2, and z in the free particle solutions and rewriting the resultant expressions in terms of the old variables x and y, one obtains the general solution for equation (3.1) in the form where A = I 1 cos(ωt) + I 3 sin(ωt) and B = I 2 cos(ωt) + I 4 sin(ωt). Here we point out that the nonlinear system (3.1) admits amplitude independent frequency of oscillations.
In the above example we have considered the new dependent variables (w 1 , w 2 ) and independent variable z to be functions of x, y and t only. We will now consider examples which admit more general transformations.
(ii) Example:2 Generalized Sundman Transformation of Type-I Let us focus our attention on the two-dimensional Mathews-Lakshmanan oscillator system of the form (Cariñena et al. 2004;Chandrasekar et al. 2009
For the case α = 0, one may note that on making a substitution y(t) = y(x(t)) into (3.6), one can obtain a non-autonomous second order ODE in y(x). Though this equation satisfies the linearization condition for point transformation (Sookmee & Meleshko 2008), finding the linearizing transformation and the general solution for the transformed ODE turns out to be nontrivial. On the other hand, we provide a straightforward procedure of linearization.
The above two integrals (3.7) can be rewritten aŝ (3.9) From the above equation, we identify the new dependent and independent variables as . (3.10) One may observe that the independent variables z 1 and z 2 are not perfect differentials even though they turn out to be identical. By utilizing the above new variables, one can transform equation (3.6), with α = 0, to the free particle equations, that is, d 2 w1 dz 2 = 0 and d 2 w2 dz 2 = 0. Unlike the earlier example, one cannot unambiguously integrate these two linear equations in terms of the original variables due to the nonlocal nature of the independent variable. To overcome this problem one should express (1 + λr 2 ) in terms of either z 1 or z 2 so that the resultant expression dz 1 = dz 2 = dt (1+λr 2 ) can be integrated. In the following, we describe a procedure to obtain an expression for the new independent variable. Now integrating equation (3.8) and using first relation in (3.10), we get where we have fixed the integration constant to be zero (without loss of generality). On the other hand from expressions (3.8) and (3.9) we get dw1 dw2 =Î 1 I2 from which one obtains whereÎ 3 is the integration constant. Equation (3.12) provides us an identity (3.13) Now squaring and adding the equations (3.11) and (3.13), we obtain ). (3.14) From (3.14), one can express (1 + λr 2 ) in terms of z 1 as .
(3.15) Substituting (3.15) in the last relation given in equation (3.10), we arrive at the following integral relationship between z 1 and t, namely, Since the variables are separated now, one can integrate this equation and obtain an expression which connects the new independent variable with the old independent variable through the relation where ω = √ λ Î2 2 (λÎ 2 3 − 1) −Î 2 1 and t 0 is the fourth integration constant. From equations (3.11)-(3.13) and (3.15), we get Substituting the expression (3.17) in the above equation (3.18) and simplifying the resultant expressions, we arrive at the following general solution for equation (3.6) with α = 0 in the form

(iii) Example:3 Generalized linearizing transformation of Type-I
In the previous example we restricted the new independent variable to be a nonlocal one and be a function of t, x and y only. Now we relax the latter condition and also allow the independent variables z 1 and z 2 to contain derivative terms, namelyẋ andẏ. To illustrate this case let us consider a two coupled second order equations of the formẍ (3.20) One can easily identify two integrals for the equation (3.20) in the form (3.21) Rewriting the above integrals as we identify the following set of linearizing transformations for the equation (3.20), that is One may note that the independent variables are not only nonlocal but also involve derivative terms. It is easy to check that equation (3.23) transforms (3.20) to the linearized form (2.3). Again, as in the previous example, one cannot directly obtain the solution for (3.20) by direct integration of the linear ODEs due to the nonlocal nature of the independent variables. This can be overcome by expressing the termẋ/y 2 in terms of either z 1 or z 2 so that the resultant equation can be integrated to obtain an explicit form for the new independent variable in terms of the old variables as we discuss below.
From (3.21), we haveẋ . (3.24) Since dw1 dz1 = I 1 (from equation (3.22)), we have w 1 = I 1 z 1 , so that where without loss of generality we have fixed the integration constant to be zero. On the other hand, from (3.22), we have dw1 dw2 = I1 I2 , which in turn gives (3.26) In terms of old variables (3.26) reads as (3.27) From the identites (3.25) and (3.27), we can express y/x interms of z 1 in the form (3.28) Now substituting (3.28) into equation (3.24), we can express (ẋ y 2 ) in terms of z 1 , and plugging the latter relation into the third relation in (3.23), we arrive at Integrating the above equation, we obtain where t 0 is the fourth integration constant. Substituting the expression z 1 = t− 1 x I1 (vide equation (3.25)) into equations (3.28) and (3.30), we obtain the general solution of (3.20) in the implicit form In the Class-A category, in all the three examples, we considered that the new independent variables, z 1 and z 2 , are identical. However, this need not always be the case in the theory of linearizing transfomations, as discussed in Sec. §2. We now present specific examples to illustrate more general transformations.

Rewriting (3.33) in the form
one can identify the new dependent and the independent variables as (3.36) One may note that now the independent variables z 1 and z 2 are not the same.
The new variables transform (3.32) to the free particle equations d 2 w1 dz 2 1 = 0 and d 2 w2 dz 2 2 = 0. From the general solutions w 1 = I 1 z 1 + I 3 and w 2 = I 2 z 2 + I 4 , where I i 's, i = 1, 2, 3, 4, are the integration constants, and using the expressions for w i 's and z i 's, i = 1, 2, given in (3.36), we arrive at the general solution for the equation (3.32) in the form where now A = I 1 sin(ω 1 t) + I 3 cos(ω 1 t) and B = I 2 cos(ω 2 t) + I 4 sin(ω 2 t).

(ii) Example:5 Mixed Point-Generalized Sundman transformation (PGST)
Let us consider the two dimensional force-free coupled Duffing-van der Pol oscillator (DVP) equation of the form x + 4(α + β(k 1 x + k 2 y) 2 )ẋ + α(3α + 4β(k 1 x + k 2 y) 2 )x = 0, y + 4(α + β(k 1 x + k 2 y) 2 )ẏ + α(3α + 4β(k 1 x + k 2 y) 2 )y = 0. (3.38) One may note that the point transformation X = k 1 x + k 2 y and Y = k 1 x − k 2 y helps one to rewrite (3.38) in a separable form, X + 4(α + βX 2 )Ẋ + α(3α + 4βX 2 )X = 0, (3.39a) Y + 4(α + βX 2 )Ẏ + α(3α + 4βX 2 )Y = 0. (3.39b) The solution to equation (3.39a) can only be obtained in implicit form (Chandrasekar et al. 2005). Consequently, equation (3.39b) cannot be solved explicitly in this way. Further, the linearization of the scalar DVP oscillator (3.39a) itself has not yet been reported. In the following we use our procedure to find the linearizing transformation and general solution to (3.38) straightforwardly. The first two integrals for (3.38) can be easily identified using the procedure given in Chandrasekar et al. (2009) in the form (3.40) The above integrals can be rewritten as from which we can obtain the following linearizing transformations, One may note that in the present problem one of the new independent variables, that is, z 2 is in a nonlocal form. In terms of the above new variables, equation (3.38) assumes the forms d 2 w1 dz 2 1 = 0 and d 2 w2 dz 2 2 = 0. Now we seek the general solution of (3.38) from the linearized equations. Integrating d 2 w1 dz 2 1 = 0, we obtain ( 3.43) where I 3 is the integration constant. Rewriting (3.43) in terms of the old variables, we get x = I 1 e −αt + I 3 y. Now substituting the relation (3.46) into the non-local variable z 2 (vide equation (3.42)), one gets (3.47) Solving the above equation, we obtain , Article submitted to Royal Society where a = 3 − 3I2 4β and t 0 is the fourth integration constant. From the expression (3.48) and equations (3.44) and (3.46) one can deduce the general solution for (3.38) in implicit form. The resultant expression coincides exactly with equation (6.18) given in Chandrasekar et al. (2009). In the present example we considered one of the independent variables to be in a nonlocal form. As we have two independent variables one can also have the possibility of having both the independent variables to be of nonlocal form. Indeed this is the case of our next example.

(iii) Example:6 Generalized Sundman transformation of Type-II (GST-II)
To illustrate the GST-II, we consider the equation of the form The first two integrals for (3.49) can be evaluated as (3.50) Rewriting these two integrals as we identify the linearizing transformations in a more general form, The GST-II, (3.52), takes equation (3.38) to the free particle equations, d 2 w1 dz 2 1 = 0 and d 2 w2 dz 2 2 = 0. To obtain the solution in terms of the original variables, we have to replace both ((x 2 + y 2 ) 2 /x 2 )dt and ((x 2 + y 2 ) 2 /y 2 )dt by the variables z 1 and t, and z 2 and t, respectively, and integrate the resultant equations.
To do so, first we rewrite the first integrals I 1 and I 2 given by (3.51) in integral forms, which in turn lead us to w 1 = I 1 z 1 and w 2 = I 2 z 2 . Since w 1 and w 2 do not contain nonlocal variables we can replace them by the old variables (vide equation (3.52)), that is where we have fixed the integration constants to be zero (without loss of generality). We observe that to integrate the last two expressions in (3.52) one should further replace z 1 and z 2 in terms of t. So we substitute the above expressions for x and y in terms of z 1 and z 2 , respectively, in the last two relations in (3.52), and obtain dz 1 = I 2 1 z 2 1 t 2 dt and dz 2 = I 2 2 z 2 2 t 2 dt. (3.54) Now integrating both the equations, we get where I 3 and I 4 are the third and forth integration constants, respectively. Plugging equation (3.55) into (3.53), we arrive at the following general solution for the equation (3.49), ( 3.56) whereÎ 3 = (I 1 I 3 − I 2 I 4 )/(I 3 1 + I 1 I 2 2 ) andÎ 4 = (I 2 I 3 + I 1 I 4 )/(I 3 2 − I 2 1 I 2 ). In the previous two examples we focussed our attention on the case in which the new independent variable(s) is(are) nonlocal and does(do) not admit velocity dependent terms. Now we relax this condition and allow either one or both the independent variables to admit velocity dependent terms but in nonlocal form.

Article submitted to Royal Society
In terms of the above new variables, equation (3.57) gets transformed to the free particle equations (2.3). One may note that one of the new independent variables is not only in non-local form but also contains derivative terms which in turn complicates the situation to obtain the general solution.
Since w 1 and z 1 are both of point transformation types, one can integrate the first free particle equation, namely, d 2 w1 dz 2 1 = 0 and obtain w 1 = I 1 z 1 + I 3 , where I 3 is an integration constant. Replacing the latter in terms of the old variables, one gets the relation (x − y) = I 1 t + I 3 . On the other hand, from the solution of the second linear equation d 2 w2 dz 2 2 = 0, we can write w 2 = I 2 z 2 ⇒ log[1 + 2λ(x + y)] = I 2 z 2 (again we assume the integration constant to be zero without loss of generality).
To evaluate z 2 , let us first substitute (3.58) into (3.61), and rewrite the latter in the form dz 2 = 2λ(ẋ +ẏ) I 2 (1 + 2λ(x + y)) dt = 2λ(I 1 + 2ẏ) I 2 (1 + 2λ(x + y)) dt. (3.62) Now subsituting the form ofẏ (vide equation (3.58)), that is, into equation (3.62), and using the relation (1 + 2λ(x + y)) = e I2z2 , we obtain (3.64) Integrating (3.64), we get where t 0 is an integration constant. Substituting the expression (3.65) into the relation 2λ(x + y) = e I2z2 − 1, we get (3.66) From equation (3.66), and the relation (x − y) = I 1 t + I 3 , we obtain the general solution for (3.57) in the form In this example we considered the case in which only one of the independent variables is in nonlocal form. Now we consider the case in which both the independent variables are in nonlocal forms. (

v) Example:8 Mixed Generalized Sundman-Generalized linearizing transformation
To illustrate this type of linearizing transformation let us consider again equation (3.6), but now with α = 0. Equation (3.6) admits the following two integrals of motion, (3.68) Rewriting these two integrals in the form and identifying the linearizing transformations, we obtain dt.