On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations
OOn linearizability via nonlocal transformations and first integrals for second-order ordinary differential equations
Dmitry I. SinelshchikovNational Research University Higher School of Economics, Moscow, RussiaSeptember 21, 2020
Abstract
Nonlinear second-order ordinary differential equations are common in various fieldsof science, such as physics, mechanics and biology. Here we provide a new family of in-tegrable second-order ordinary differential equations by considering the general case ofa linearization problem via certain nonlocal transformations. In addition, we show thateach equation from the linearizable family admits a transcendental first integral and studyparticular cases when this first integral is autonomous or rational. Thus, as a byproductof solving this linearization problem we obtain a classification of second-order differentialequations admitting a certain transcendental first integral. To demonstrate effectivenessof our approach, we consider several examples of autonomous and non-autonomous sec-ond order differential equations, including generalizations of the Duffing and Van der Poloscillators, and construct their first integrals and general solutions. We also show thatthe corresponding first integrals can be used for finding periodic solutions, including limitcycles, of the considered equations.
Here we consider the following family of non-autonomous nonlinear second-order differentialequations y zz + f ( z, y ) y z + g ( z, y ) y z + h ( z, y ) = 0 , (1.1)where f , g and h are arbitrary sufficiently smooth functions. We assume that gh (cid:54)≡ and that f + g y + h yy (cid:54) = 0 , i.e. we exclude the linear subcase of (1.1) from the consideration.Equations from family (1.1) often appear in numerous applications in mechanics, physicsand so on [1, 2]. Therefore, various aspects of integrability of (1.1) have been studied in anumber of works (see, e.g., [3–17]). For example, in [5, 7, 9, 10] authors considered applicationsof several linearizing transformations and λ -symmetries for finding first integrals of equationsfrom family (1.1). In particular, in [5] it was shown that equations admitting a linear withrespect to the first derivative first integral form exactly the same class as equations linearizedto the Laguerre normal form of linear second order differential equations (the latter class wasobtained in [3]). Authors of [5] also demonstrated that equations from the corresponding classpossess a certain λ -symmetry and there is a subclass of completely integrable equations withtwo independent first integrals. In [6–9] various connections between linearizability of second-order differential equations and the existence of certain first integrals, in particular rationalones, were studied. Authors of [11, 12] applied the Jacobi last multiplier approach for studyingintegrability of (1.1), while in [13, 14] equivalence problems via point transformations were1 a r X i v : . [ n li n . S I] S e p tudied. Connections via nonlocal transformations between equations from (1.1) and variousPainlev´e type equations were considered in [18–21].Here we deal with the linearization problem for (1.1) via the generalized Sundman trans-formations, which have the form w = F ( z, y ) , dζ = G ( z, y ) dz, (1.2)where F and G are some sufficiently smooth functions satisfying F y G (cid:54) = 0 . This problem waspreviously studied in [3, 4]. While in [3] linearization to the Laguerre normal form of secondorder linear differential equation, namely to the equation w ζζ = 0 , was considered, in [4] it wasshown that for the linearization via transformations (1.2) it is insufficient to use the Laguerrenormal form of a linear second order differential equation and connections between (1.1) and w ζζ + βw ζ + αw = 0 , (1.3)were considered. Here α (cid:54) = 0 , β (cid:54) = 0 are arbitrary parameters. Although authors of [4] studiedthe equivalence problem between (1.1) and (1.3) via (1.2), only a particular case of transfor-mations (1.2), specifically the case of F z = 0 , was considered. However, it is known that thereare interesting from an applied point of view nonlinear oscillators that can be linearized via(1.2) only if F z (cid:54) = 0 (see, e.g. [22]). Therefore, in this work we consider the full linearizationproblem for (1.1) and find all equations from family (1.1) that can be linearized with the help of(1.2) with F z (cid:54) = 0 . We demonstrate that there are nontrivial examples of equations from (1.1)that can be linearized only via (1.2) with F z (cid:54) = 0 . Furthermore, we show that each linearizableequation from (1.1) admits a certain first integral, which can be explicitly constructed via theparameters of the studied equation and linearizing transformations. This follows from the factthat linear equation (1.3) possesses an autonomous first integral and we believe that this is thefirst time when the corresponding first integrals are obtained for linearizable equations from(1.1). We also separately consider the cases when this first integral is autonomous or a ratio-nal/polynomial function. Finally, let us remark that authors of [4] also included a constantparameter γ in (1.3), but it can be easily removed via the transformation F → F + γ/α , and,consequently, we do not take it into consideration.Notice also that the linearization problem for family (1.1) via a more general class of nonlocaltransformations, when the function G in (1.2) depends on y z was considered (see, e.g. [7, 8, 23]and references therein). For instance, in [8] linearization problem for (1.1) via (1.2) with G ( z, y, y z ) = G ( z, y ) y z + G ( z, y ) was studied. Authors of [8] showed that equations forthis linearizable class possess a certain rational first integral and a λ -symmetry, which can becalculated in terms of the coefficients of the corresponding equation.The rest of this work is organized as follows. In the next Section we present the equiva-lence criterion for (1.1) and (1.3). We also show how to construct a first integral for linearizableequation from (1.1) and present several interesting subcases of linearizable equations from (1.1),namely Darboux integrable cases and equations with rational non-autonomous first integrals.In Section 3 we provide several examples of linearizable equations from (1.1) including para-metrically forced generalizations of the Duffing and Van der Pol equations. In the last sectionwe briefly discuss and summarize our results. Let us start with some preliminary results. First we introduce a canonical form of (1.1) withrespect to (1.2).
Proposition 2.1
Family of equations (1.1) is closed with respect to (1.2) and its canonicalform is y zz + g ( z, y ) y z + h ( z, y ) = 0 . (2.1)2 roof. The closedness of (1.1) with respect to (1.2) can be checked by direct calculations.Thus, without loss of generality, one can assume that f ( z, y ) = 0 . Indeed, substituting thetransformation ˜ y = (cid:82) exp { f } dy , which is a particular case of (1.2), into (1.1) we get ˜ y zz + ˜ g ( z, ˜ y )˜ y z + ˜ h ( z, ˜ y ) = 0 , (2.2)where f = (cid:90) f dy, ˜ g = g − f z , ˜ h = e f h + (2 f z − g ) (cid:90) f z e f dy − (cid:90) ( f zz + f z )e f dy. (2.3)In order to obtain results for family of equations (1.1) from the results for (2.2) we need tomake the following substitutions y → (cid:90) e f dy, g → g + 2 f z ,h → e − f (cid:18) h + (cid:90) f z e f dyg + (cid:90) ( f zz + f z )e f dy (cid:19) . (2.4)This completes the proof. (cid:50) Consequently, further we assume that f ( z, y ) = 0 and study equivalence problem for (2.1).Now let us show that (1.3) has an autonomous first integral that can be used for constructingfirst integrals for linearizable equations from (2.1). Indeed, it is easy to verify that the followingexpression I = (2 w ζ + ( β + ρ ) w ) ρ + β (2 w ζ + ( β − ρ ) w ) ρ − β , (2.5)where ρ = (cid:112) β − α (cid:54) = 0 , is a first integral of (1.3). If ρ = 0 , instead of (2.5) one needs to use I = (2 w ζ + βw ) exp (cid:26) βw w ζ + βw (cid:27) . (2.6)Notice also that if ρ is imaginary, i.e. β − α < , first integral (2.5) can be transformed intoa real form as follows I = ln (cid:8) αw + βww ζ + w ζ (cid:9) − β (cid:112) − ρ arctan (cid:40) w ζ + βw (cid:112) − ρ w (cid:41) . (2.7) Remark 2.1
Notice that from the results of [5] it follows that (1.3) possesses two functionallyindependent first integrals, which are linear functions with respect to w ζ . These integrals are I = e β + ρ ζ (2 w ζ + ( β − ρ ) w ) , I = e β − ρ ζ (2 w ζ + ( β + ρ ) w ) . (2.8) However, (2.8) cannot be used for constructing first integrals of linearizable equations from (2.1) , since transformations (1.2) do not map a non-autonomous first integral of (1.3) into afirst integral of a linearizable equation from (2.1) .On the other hand, integral (2.5) can be easily obtained from (2.8) as I = I ρ − β I ρ + β . Ifone considers another function of I and I that gives an autonomous first integral of (1.3) ,one obtains an integral, that is a function of (2.5) , since (1.3) can admit at most one, upto a functional dependence, autonomous first integral. In other words, any autonomous firstintegral of (1.3) is a function of (2.5) . Thus, only (2.5) (or any function of it) can be used forconstructing first integrals for linearizable equations from (2.1) . The cases of β − α < and β − α = 0 can be treated in a similar way. w , w ζ and w ζζ via F and G into (1.3). This yields to y zz + gy z + h = 0 , (2.9)provided that GF yy − F y G y = 0 , (2.10)holds. Here g = 2 GF yz − F y G z − F z G y + βG F y GF y , h = GF zz − F z G z + βG F z + αG FGF y . (2.11)Therefore, equation (2.1) can be transformed into (1.3) if it is of the form (2.9) and (2.10)holds.Conversely, if the functions F and G satisfy (2.10), (2.11) then equation can be trans-formed into (1.3) with the help of (1.2). As a result, compatibility conditions for the followingoverdetermined system of partial differential equations for the functions F and GGF yy − G y F y = 0 ,gGF y − GF yz + F y G z + F z G y − βG F y = 0 ,hGF y + G z F z − GF zz − αG F − βG F z = 0 . (2.12)give us the necessary and sufficient conditions for (2.1) to be equivalent to (1.3) via (1.2).Now our goal is to explicitly find correlations on the functions f and g that provide com-patibility of (2.12) and, hence, define equations of the form (2.1) that can be both linearizableand admit a certain first integral. Although direct computation of the compatibility conditionsfor (2.12) is quite cumbersome, we can considerably simplify this system, which allows us toexplicitly find required compatibility conditions.Solving the first equation from (2.12) we get that G = AF y , (2.13)where A = A ( z ) (cid:54)≡ is an arbitrary sufficiently smooth function. With the help of this relation,from (2.12) we obtain βAF y − g − A z A − F z F yy F y + F yz F y = 0 ,αA F F y + βAF z − h − (cid:18) A z A + F yz F y (cid:19) F z F y + F zz F y = 0 . (2.14)The first equation from (2.14) can be integrated once with respect to y . As a result, we obtain F z + (cid:18) βAF − m − C − A z A y (cid:19) F y = 0 , (2.15)where m y = g and C ( z ) is an arbitrary sufficiently smooth function.With the help of (2.15), from the second equation from (2.14) we get F F y + 1 αA (cid:18) m z + C z − h + A zz A y − A z A m + A z A C − A z A y (cid:19) = 0 . (2.16)4ntroducing in (2.15), (2.16) the following notations p = m + C + A z A y, l = β αA (cid:18) A z A p + h − p z (cid:19) , L = βF, (2.17)we have LL y − l = 0 , LL z + AlL − lp = 0 . (2.18)This system is quite simple in comparison with (2.12). If we consider (2.18) as an overdeter-mined system for the function L , the corresponding compatibility conditions give the necessaryand sufficient conditions for linearization of (2.1) via (1.2) provided that one takes into accountnotations (2.17).The compatibility conditions for (2.18) split into four separate cases: the generic case andthree particular cases. During the computation of the compatibility conditions we assume that L (cid:54) = 0 , l (cid:54) = 0 and A (cid:54) = 0 since otherwise transformations (1.2) degenerate. To simplify furtherrepresentation we introduce the following notations P = l yy , Q = p yy , R = pl y + lp y − l z . (2.19)The generic case of the compatibility conditions is A P l R y R − P A l + (cid:0) A l l y − A R l l y + R (cid:1) P −− A l y l R y + l y R (cid:0) A l l y − R (cid:1) R y − A l y l (cid:0) A l l y − R (cid:1) = 0 ,A P Ql + (cid:0) AR lp y − A Rl l y − A l R y + ARlpR y − AR −− ARlR z + R lA z (cid:1) P − Al y l (cid:0) A l l y − R (cid:1) Q − l y ( l y R − lR y ) (cid:0) A ll y −− ARp y − ApR y + AR z − RA z ) = 0 ,l (cid:0) A Rl l y − A l R y − R (cid:1) P y − A P l R + A P l R yy − ll y (cid:0) A l l y − R (cid:1) R yy −− (cid:0) A l l y + 3 R (cid:1) ( Rl y − lR y ) P − l y ( Rl y − lR y ) (cid:0) A ll y − RR y (cid:1) = 0 , (2.20)while the function L is given by L = Al (cid:0) lR y l y − P Rl − Rl y (cid:1) A P l l y − A ll y − P R + RR y l y . (2.21)Let us briefly describe the process of computation of conditions (2.20) and expression (2.21).We consider (2.18) as an overdetermined system of equations for L and apply the Riquier–Janetcompatibility theory (see, e.g. [24]) for computing the corresponding compatibility conditions.This is done via calculating various mixed partial derivatives of L with respect to z and y and comparing them. The comparison of L yz and L zy leads to the expression for L via p and l . Then, with the help of this expression and expressions for Lzyy , Lyyz and
Lyzy wefind expression (2.21) and the first condition from (2.20). Further computing and comparingthird order mixed derivatives of L we obtain the second condition from (2.20). Finally, withthe help of expressions for L yyyz , L zyyy and L yyzy we find the last compatibility condition from(2.20). Computation of further mixed partial derivatives of L does not lead to new compatibilityconditions. In addition, to verify that all compatibility conditions are obtained we compareour results with those produced by the Rif package [24]. Our results and results produced byRif coincide. Let us remark that further we do not provide details of the computation of thecompatibility conditions since they are similar to those given above.5ow we need to consider particular cases of the compatibility conditions. First, we deal withthe case when the denominator of (2.21) vanishes. As a result, we get the following relations A P l − A l l y + R l y = 0 ,l (cid:0) P lp − pl y + 5 Rl y − lR y (cid:1) A − R ( R − pl y ) = 0 , ( AP p + AQl + 2 Al y p y − AR y + A z l y ) L − R z L ++ l (2 Alp y + 2 Apl y − AR + lA z ) L + l (cid:0) A l + Rp (cid:1) L − Apl = 0 , l RA (cid:0) A l + 5 Rp (cid:1) l y + 5 Ql y A l − l R QA − A l Q y + 2 R −− R (cid:0) − A Rl p y + A l pR y + 6 A R l + A l R z − ARl A z + 2 R p (cid:1) l y = 0 , (2 Al zzy + 4 A z l z,y + 2 l y A zz ) L + (cid:0) l zzz − A P lp − A pl y − lp zzy −− pl zzy − l y p zz − l z p zy − p z l zy − p y l zz ) L + (cid:0) AP lp + AQl p + 3 Alpl y p y ++ Ap l y − Al p zy + 5 Alpl zy − All y p z − All z p y − Apl y l z −− l A z p y + 8 lpA z l y + 8 All zz + 6 Al z + 2 l A zz + 10 lA z l z (cid:1) L −− l (cid:0) A l p y + A lpl y − A ll z − Al A z + 6 lpp zy + 6 p l zy + 6 pl y p z ++6 pl z p y − pl zz ) L − l (cid:0) Alpp y − A l + 3 Ap l y + 2 Alp z − Apl z − lpA z (cid:1) = 0 , (2.22)and L is given by L = Al (cid:16) A l (cid:0) A l − Rp (cid:1) l y − RQl A + ( pR y + R z − Rp y ) l y A l ++5 A l l y R − Al l y A z R + R (2 pl y − R ) (cid:17)(cid:16) Ql y A l − A l R (cid:0) A l + 5 Rp (cid:1) l y −− A l QR + Rl y A l ( pR y + R z − Rp y ) + 6 A l l y R − Al l y A z R + R (2 pl y − R ) (cid:17) − . (2.23)The next case corresponds to the vanishing of the denominator of (2.23). Consequently, weget that l (cid:0) P l − l y (cid:1) A + l y R = 0 , l (cid:0) Ql − Rl y (cid:1) A + R = 0 ,l (cid:0) P lp + Ql − pl y + Rl y − lR y (cid:1) A + l y pR = 0 ,l (cid:0) P lp + Ql p − p l y + 2 Rlp y + 6 Rpl y − lpR y − R − lR z (cid:1) A −− l y A l + A z Al R + pR ( pl y − R ) = 0 , (2.24)while L satisfies the equation L − RAl y + l l y = 0 . (2.25)Finally, in the case of l y = 0 we obtain l y = 0 , l z − p y l z l + 3 p y l z l − l (cid:0) A Ql + p y (cid:1) = 0 ,pl z − (cid:0) A l + 3 lpp y (cid:1) l z + (cid:0) l pp y − A l p y − Al A z (cid:1) l z ++ l (cid:0) lp y − lp zy + l zz (cid:1) A + A z p y Al − p y pl = 0 , (2.26)and L = Al lp y − l z . (2.27)We do not need to consider the case of lp y − l z = 0 separately, since it results in eitherdegeneration of transformations (1.2) or reduces to subcases of (2.22) or (2.24).The above results can be summarized as follows:6 heorem 2.1 Equation (2.1) can be transformed into (1.3) if and only if one of the sets ofcorrelations (2.20) , (2.22) , (2.24) or (2.26) holds. Remark 2.2
In order to check compatibility conditions for a particular member of (2.1) oneneeds to calculate the values of the functions p and l via g and h with the help of the relations (2.17) taking into account that m y = g . Then, one needs to substitute the corresponding valuesof the functions p and l into one of the sets of the compatibility conditions (2.20) , (2.22) , (2.24) or (2.26) and check whether they hold at some values of A (cid:54) = 0 and C . We present a detailedalgorithm for verifying compatibility conditions at the beginning of the next section. As an immediate consequence of Theorem 2.1 we get
Corollary 2.1
If one of the sets of correlations (2.20) , (2.22) , (2.24) or (2.26) holds thenequation (2.1) admits the following first integral I = A − ρ (cid:18) y z + 2 p − β − ρβ AL (cid:19) ρ + β (cid:18) y z + 2 p − β + ρβ AL (cid:19) ρ − β , (2.28) where p is given in (2.17) , ρ = (cid:112) β − α (cid:54) = 0 and L = βF . If ρ = 0 then the first integral is I = 2 y z + 2 p − ALA exp (cid:26) AL y z + 2 p − AL (cid:27) . (2.29)It is interesting to understand when transformations (1.2) keep first integral (2.5) au-tonomous. One can show that this is true if and only if G z = F z = 0 . As a consequence,we have that the following statement holds Corollary 2.2
Equation of the form y zz + g ( y ) y z + h ( y ) = 0 , is integrable with the first integral I = (2 βy z + ( ρ + β )( m + µ )) ρ + β (2 βy z + ( ρ − β )( m + µ )) ρ − β , if β ( hg y − gh y ) + αg = 0 , where m y = g . Let us also consider the case when transcendental first integral (2.28) becomes a rationalone. One can show that this is true if the following relation holds α = (1 − r ) β , where r (cid:54) = 0 is a rational number. As a consequence, we have that Corollary 2.3
If one of the sets of correlations (2.20) , (2.22) , (2.24) or (2.26) holds and α = (1 − r ) β, r = nk , k, n ∈ Z \ { } , (2.30) then equation (2.1) admits the following rational first integral I = A − n (2 y z + 2 p − (1 − r ) AL ) n + k (2 y z + 2 p − (1 + r ) AL ) n − k , (2.31) where p is given in (2.17) and L = βF . Thus, in this section we have explicitly find correlations on functions g and h that give us thelinearization criterion for (2.1) via generalized Sundman transformations. We have also showedthat once an equation from (1.1) is linearizable it possesses a certain first integral. Moreover,we have isolated linearizable families of equations that admits an autonomous first integral ora rational one. 7 Examples
In this section we provide several new examples of linearizable equations of form (1.1). First,we demonstrate that there are indeed equations from family (1.1) with coefficient satisfyingconditions from Theorem 2.1, but which cannot be linearized via (1.2) with F z = 0 . Then,we provide several example of both non-autonomous and autonomous nonlinear oscillatorsincluding generalizations of the Duffing and Van der Pol oscillators, that can be linearized via(1.2) with F z (cid:54) = 0 .Let us present an algorithm for verifying that a particular member of (2.1) can be linearizedwith the help of (1.2). It consists of the following three steps. First, using (2.17) and taking intoaccount that m y = g we calculate the values of the functions p and l via g and h . Second, wesubstitute the corresponding values of the functions p and l into one of compatibility conditions(2.20), (2.22), (2.24) or (2.26). As a result of this substitution, we obtain polynomials in y ,whose coefficients are functions of z . Equating coefficients of these polynomials to zero, weget a system of equations for the functions A and C . If this system is satisfied for any valuesof A (cid:54) = 0 and C , then the corresponding equation from (2.1) is linearizable. Third, if oneof the sets of the compatibility conditions is satisfied, we calculate the value of L via one ofthe relations (2.21), (2.23), (2.25), (2.27) and then it is easy to find the explicit form of thelinearizing transformations with the help of (2.13) and (2.17). Example 1 . Let us consider the following equation from family (2.1) y zz + ( βe − δz y − δ ) y z + e − δz y ( αe − δz y − βδ ) = 0 . (3.1)In order to check that this equation can be linearized via (1.2) we use the algorithm pre-sented above. With the help of (2.17) we find that p = βe − δz / y − δy − βδ e δz /α and l = 2 β e − δz y ( αe − δz y − δ ) /α . Substituting these values of p and l into (2.20) and equatingcoefficients at the same powers of y we find that A = e δz / and C = − βδ e δz /α and from (2.21),(2.13) and (2.17) we get that L = β ( e − δz y − δ /α ) , F = e − δz y − δ /α and G = e − δz y . Asa result, we have that (3.1) can be linearized via (1.2) and its general solution can be presentedin the following parametric form y = ± e δz (cid:18) w + 2 δ α (cid:19) / , z = ± (cid:90) dζ (cid:0) w + δ α (cid:1) / , (3.2)where w is the general solution of (1.3).From Corollary 2.1 it follows that (3.1) possesses the first integral I = e − ρδz (cid:18) y z − δy + βe − δz y − βδ α e δz − e δz α ( β − ρ )( αe − δz y − δ ) (cid:19) ρ + β (cid:18) y z − δy + βe − δz y − βδ α e δz − e δz α ( β + ρ )( αe − δz y − δ ) (cid:19) ρ − β , (3.3)if ρ (cid:54) = 0 and the first integral I = e δz (cid:18) y z − δy + β y e δz − δ β e δz (cid:19) exp (cid:26) δ e δz − β y δ e δz − β y + 4 βδe δz y − βe δz y z (cid:27) , (3.4)if ρ = 0 .Equation (3.1) can be considered as a non-autonomous generalization of the damped Duffingoscillator. Notice that one can show that equation (3.1) cannot be linearized via (1.2) with F z = 0 and possesses only one Lie point symmetry. Therefore, (3.1) provides an example of8igure 1: Projections of (3.9) on the plane z = c for different values of c : s ( z ) = sin z , α = − β = 1 (left figure); s ( z ) = e − z , α = 10 , β = − (middle figure); s ( z ) = sin z cos( πz ) , α = 10 , β = 1 (right figure).an equation that cannot be both integrated with the help of the classical Lie approach andlinearized with the help of the restricted case of transformations (1.2). Example 2.
Consider a family of parametrically forced Duffing oscillators with lineardamping y zz + ( b y + b ( z )) y z + a y + a ( z ) y + a ( z ) y = 0 , (3.5)where b (cid:54) = 0 and a (cid:54) = 0 are certain parameters and b , a and a are certain functions of z .Now we need to check whether coefficients of (3.5) satisfy one of the sets of the compatibilityconditions. For the sake of simplicity, we assume that C = 0 . The case of C (cid:54) = 0 can be treatedin the same way.According to the algorithm presented above, at the first step we find that p = A z A y + b y b y,l = β αA (cid:2) a y + a y − b ,z y + a y ) A + ( b y + 2 b y ) A z − AA zz y + 4 A z y (cid:3) . (3.6)Substituting (3.6) into (2.20) and collecting coefficients at the same powers of y , we find that if b = 2 β, b = 3 s, a = 2 α, a = 2 βs, a = 2 s + s z , A ( z ) = e − (cid:82) s ( z ) dz , (3.7)then conditions (2.20) are satisfied. Here s = s ( z ) is an arbitrary function. As a consequence,with the help of (2.21), (2.17) and (2.13), we get that F = e (cid:82) sdz y and G = 2 y .As a result, we have that the equation y zz + (2 βy + 3 s ) y z + 2 αy + 2 βsy + (2 s + s z ) y = 0 , (3.8)can be linearized with the help of (1.2).From Corollary 2.1 it follows that (3.8) has the following first integral if ρ (cid:54) = 0 I = e ρ (cid:82) sdz (cid:0) y z + 2 sy + ( β + ρ ) y (cid:1) ρ + β (cid:0) y z + 2 sy + ( β − ρ ) y (cid:1) ρ − β , (3.9)and if ρ = 0 this first integral is I = e (cid:82) sdz (cid:0) y z + 2 sy + βy (cid:1) exp (cid:26) βy y z + 2 sy + βy (cid:27) . (3.10)The general solution of (3.8) can be presented as follows y = ±√ w e (cid:82) sdz , (3.11)9igure 2: Plots of one parametric families of solutions of (3.8) for different forcing functions: s ( z ) = z/ ( z + 1) , α = 4 , β = 5 (left figure); s ( z ) = 2 z , α = 4 , β = 5 (middle figure); s ( z ) = tan z , α = 3 . , β = 4 (right figure).where z is given by ± (cid:90) dζ √ w = (cid:90) e − (cid:82) sdz dz, (3.12)and w is the general solution of (1.3).Let us discuss some properties of solutions of (3.8). In Fig.1 we demonstrate projections of(3.9) on the plane z = const for different values the forcing function s and other parameters.Notice that for the left figure the integration constant corresponds to y (0) = 1 , y z (0) = 0 andfor the other cases the integration constant corresponds to y (0) = 1 / , y z (0) = 1 / . One cansee that equation (3.8) has various types of periodic solutions even if the forcing function is notperiodic. Furthermore, one can show by varying the integration constant that these periodictrajectories are not isolated in the phase space, namely, they are not limit cycles.One-parametric families of solutions of (3.8) can be easily obtained from (3.9) and (3.10)as follows y = 2 exp {− (cid:82) s ( z ) dz } ( β ± ρ ) (cid:82) exp {− (cid:82) s ( z ) dz } dz + C , (3.13)where C is an arbitrary constant. We demonstrate plots of (3.13) for different forcing functionsand values of parameters in Fig.2. One can see that depending on the forcing functions thesesolutions may be solitary or periodic waves. Example 3.
Let us consider the following family of non-autonomous nonlinear oscillators y zz + (cid:0) b y + b (cid:1) y z + a y + a y + a y + a y = 0 , (3.14)where b i = b i ( z ) , i = 1 , and a j = a j ( z ) , j = 1 , , , are some functions and b , a (cid:54)≡ .Equation (3.14) can be considered as a parametrically forced φ –Van der Pol oscillator or as aparametrically forced extended Duffing–Van der Pol system (see, e.g. [25, 26]).Now we find a case of (3.14), whose coefficients satisfy (2.20). First, we compute the valuesof l and p p = A z yA + b y b y + C, l = β αA (cid:104) ( b y + 3 b y + 3 C ) AA z ++3(2 A z − AA zz ) y + (3 a y + (3 a − b ,z ) y + 3 a y + 3( a − b ,z ) y − C z ) A (cid:105) . (3.15)Second, we substitute these values into (2.20) and equate coefficients at the same powers of y .10igure 3: Projections of (3.20) on the plane y = c for different values of c (left figure) and twonumerical solutions of the Cauchy problem for (3.17) (right figure) at α = 4 , β = 5 , µ = − and s ( z ) = e sin z (right figure).As a consequence, we obtain that b = 3 βs, b = 5 s z s , a = 3 αs , a = 2 βs z , a = 3 µ,a = 2 s zz s , A = 1 /s, C = βµ/ ( αs ) . (3.16)Here s ( z ) (cid:54)≡ is an arbitrary sufficiently smooth function and µ is an arbitrary parameter.Third, using (3.16), (2.21), (2.13) and (2.17) we get that the equation y zz + (cid:18) βsy + 5 s z s (cid:19) y z + 3 αs y + 2 βs z y + 3 µy + 2 s zz s y = 0 , (3.17)can be linearized via transformations (1.2) with F = s y + µα , G = 3 sy . (3.18)Notice that the general solution of (3.17) in a nonlocal form can be obtained by inverting (1.2)with (3.18) as follows y = 1 s (cid:16) w − µα (cid:17) , (cid:90) sdz = (cid:90) dζ y , (3.19)where w is the general solution of (1.3).With the help of 2.1 we find that (3.17) possesses a first integral I = (cid:0) αsy z + 4 αys z + 3( β + ρ )( αs y + µ ) (cid:1) ρ + β (cid:0) αsy z + 4 αys z + 3( β − ρ )( αs y + µ ) (cid:1) ρ − β , (3.20)if ρ (cid:54) = 0 , and a first integral I = (6 βsy z + 4 βs z y + 3 β s y + 12 µ ) exp (cid:26) β s y + 12 µ βsy z + 4 βs z y + 3 β s y + 12 µ (cid:27) , (3.21)if ρ = 0 . 11ow we discuss some properties of integrals (3.20) and (3.21). In the left part of Fig. 3 wedemonstrate projections of (3.20) at certain values of the parameters on the y = const planefor different values of this constant. One can see that there are periodic trajectories admittedby (3.17). We argue that these periodic trajectories are limit cycles. To support this claimin the right part of Fig. 3 we demonstrate the results of numerical solution of the Cauchyproblem for (3.17). We see that nearby trajectories in the phase space converge to a certainclosed trajectory, and, thus, there is indeed a limit cycle in (3.17). One can also observe asimilar situation for first integral (3.21). Finally, if we assume that the forcing function s ( z ) isperiodic and has no zeros on the real line, one can again find a limit cycle in (3.17). Example 4.
Now we consider the following equation from family (1.1) y zz + y z y + (cid:18) µ + βy (cid:19) y z − µ y + βµy − αy = 0 , (3.22)where µ (cid:54) = 0 is an arbitrary parameter. If we transform (3.22) into its canonical form via y → √ y , one can verify with the help of the above proposed algorithm, that the coefficients ofthe corresponding equation of type (2.1) satisfy conditions (2.20). Indeed, in this case we havethat p = 2 µy − β/ √ y , l = − β e − µz / (4 y ) and A = − e − µz satisfy (2.20) and L = β e − µz / √ y .Consequently, equation (3.22) can be linearized via (1.2) with F = e − µz y , G = 1 y . (3.23)The general solution of (3.22) can be expressed in the parametric form as follows y = e − µz w , z = 13 µ ln (cid:26) µ (cid:90) dζw (cid:27) . (3.24)With the help of Corollary 2.1 we find that (3.22) has the first integral I = (cid:18) e − µz y ( y z + µy ) − ( β + ρ ) e − µz y (cid:19) ρ + β (cid:18) e − µz y ( y z + µy ) − ( β − ρ ) e − µz y (cid:19) ρ − β , (3.25)if ρ (cid:54) = 0 and the first integral I = e − µz (cid:18) y ( y z + µy ) − βy (cid:19) exp (cid:26) ββ − µy − y y z (cid:27) , (3.26)if ρ = 0 . Notice that equation (3.22) provides an example of an autonomous nonlinear oscil-lator from family (1.1), i.e. an equation with quadratic nonlinearity with respect to the firstderivative, that can be linearized via (1.2) with F z (cid:54) = 0 .In this section we have provided several examples of linearizable equations from family(1.1) that can be transformed into (2.2) via (1.2) only if F z = 0 . We have also showed thatthe corresponding first integrals allows us to find periodic trajectories. including limit cycles,admitted by the considered nonlinear oscillators. In this work we have considered family (1.1) of nonlinear second order ordinary differentialequations. We have studied the complete linearization problem for this family of equations viathe generalized Sundman transformations and obtained linearizability conditions in the explicitform. We have also shown that each linearizable equation from (1.1) admits a certain transcen-dental first integral. As a consequence, we classify all equations of form (1.1) that possess this12ranscendental first integral. We have also separated families of equations with autonomousand rational first integral. We have provided several nontrivial examples of applications of thelinearizing transformations including generalizations of the Duffing and Van der Pol oscillators.In particular, we have demonstrated that our approach can be used for finding centers and limitcycles admitted by equations from the considered family.
This research was supported by Russian Science Foundation grant No. 19-71-10003. Numericalcalculations in Section 3 were supported by Russian Science Foundation grant No. 19-71-10048.
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