Hypergeometric First Integrals of the Duffing and van der Pol Oscillators
HHypergeometric First Integrals of the Duffingand van der Pol Oscillators
Tomasz Stachowiak * Department of Applied Mathematics and Physics,Graduate School of Informatics, Kyoto University,606-8501 Kyoto, Japan
Abstract
The autonomous Duffing oscillator, and its van der Pol modification, areknown to admit time-dependent first integrals for specific values of parameters.This corresponds to the existence of Darboux polynomials, and in fact more canbe shown: that there exist Liouvillian first integrals which do not depend ontime. They can be expressed in terms of the Gauss and Kummer hypergeometricfunctions, and are neither analytic, algebraic nor meromorphic. A criterion forthis to happen in a general dynamical system is formulated as well. The Duffing and the van der Pol oscillators are among the simplest (at least in form)dynamical systems, present in biology [1, 2], nano-electronics and mechanics [3] aswell as many subfields of physics [4, 5] (see the last one for a more comprehensivelist of references). The systems’ integrability, or solvability, is still an active topicwith only partial results available [6, 7, 8] and the aim of the present article is touse them as examples of how non-analytic Liouvillian first integrals can arise inpolynomial differential equations. This is the content of Lemma 1 in Section 2 andits generalization in Section 5.The force-free form of the basic Duffing oscillator is¨ u + ˙ u + ω u + u =
0, (1) * [email protected] a r X i v : . [ m a t h - ph ] N ov r, more generally, ¨ u + α ˙ u + ω u + γ u n =
0. (2)With a suitable rescaling of u and the time, α and γ can be made equal to 1, so theonly essential parameters will be ω and n . Although the case of immediate interestis that of integer n >
1, the extension to real n is straightforward if u n is replaced by | u | n − u .For the specific value of ω = , explicit solutions of (1) in terms of Jacobianelliptic functions were found in [6]. The authors note that the equation admits atransformation W = (cid:112) t /3 u ; Z = −(cid:112) − t /3 , (3)which turns the equation into a solvable one W (cid:48)(cid:48) ( Z ) = − W . (4)It can be regarded as Hamiltonian with H = W (cid:48) + W , (5)which is then conserved, but upon going back to u and ˙ u , the first integral defined by H becomes time dependent: I D = e t (cid:181)
12 ˙ u + u ˙ u + u + u (cid:182) . (6)A generalization of the above approach, which utilizes transformations of the form (3),was formulated in [7], and gives a more constructive method of finding first integrals.As the solutions are explicitly available, this time dependence of I D does not seemto be a practical problem – one could argue that a first integral can be obtained(at least locally) by elimination of time between the solutions, but that is seldompractically computable. Thus, the question emerges of whether a “proper” time-independent first integral exists. The goal of this investigation is to show that thisis indeed the case for the general unforced Duffing oscillator, and partly so for theDuffing-van der Pol oscillator. What is more, the existence of such an integral can beascertained without the a priori knowledge of the solutions.The first main result of the present letter is the following Theorem 1.
If the frequency ω of the Duffing oscillator (2) with general nonlinearterm of degree n satisfies ω = n + n + , (7)2 hen the equation has a Liouvillian first integral. When n is odd, in the half-planesseparated by u + (3 + n ) ˙ u = the integral is given byI = V − n + (cid:183) V + ( n − ω ς u F (cid:181)
12 , 1 n + n + n + u n + V (cid:182)(cid:184) , (8) where V = u n + + (2 u + ω ( n +
3) ˙ u ) and ς = sign(2 u + ( n +
3) ˙ u ) ; alternatively, in thehalf-planes separated by u = it isI = V − n + − n − n + n − n + ω ς (cid:48) u + ( n +
3) ˙ u (cid:112) V F (cid:181)
12 , nn + − u n + V (cid:182) , (9) with ς (cid:48) = sign( u ) and V as above. Each F is real when its argument is less than 1.When n is even, so that V = defines a real invariant curve C , the above expressionsremain valid if I is taken to the right of C , and I to its left with ς (cid:48) = ( − n /( n + . The second equation of interest is the Duffing-van der Pol oscillator, which exhibitsthe same type of time-dependent first integrals. The equation is¨ u + (1 + β u m ) ˙ u + ω u + u n =
0, (10)and it was shown in [8] to admit a whole parametric family of time-dependentintegrals. The required condition is that m = n − n + β ω = n β , and theconserved quantity is I P = e n β t (cid:181) ˙ u + β − n β u + β n u n (cid:182) . (11)Similarly to the previous system, this quantity can be used to obtain a time indepen-dent first integral, leading to the second result: Theorem 2.
If the parameters of the Duffing-van der Pol oscillator (10) with generalnonlinear term of degree n satisfy ω = n ( n + , β = n + and m = n −
1, (12) then the equation has a Liouvillian first integral given byI = (cid:175)(cid:175) u n + ω ( u + β ˙ u ) (cid:175)(cid:175) − n (cid:34) + ( n − uu + β ˙ u F (cid:195)
1, 1, 1 + n , − u n / ω u + β ˙ u (cid:33)(cid:35) , (13) which is real when − ω − u n /( u + β ˙ u ) < ; or alternatively byI = (cid:175)(cid:175) u n + ω ( u + β ˙ u ) (cid:175)(cid:175) − n (cid:34) − uu + β ˙ u F (cid:195)
1, 1, 2 − n , 1 + u n / ω u + β ˙ u (cid:33)(cid:35) , (14)3 hich is real when ω − u n /( u + β ˙ u ) < . Additionally, the singularities of the hypergeo-metric function correspond to the invariant sets ω ( u + β ˙ u ) + u n = and u + β ˙ u = . The above system can be further generalized to¨ u + (1 + β u n − ) ˙ u + ω u + u n + ϕ u n − =
0, (15)which was analyzed with the Prelle-Singer procedure in [9], and with the Lie sym-metry method in [10]. It differs from the first two in that the specific structure of itsinvariant sets makes the hypergeometric first integral confluent:
Theorem 3.
If the parameters of the generalized Duffing-van der Pol oscillator (15) satisfy ω = n ( n + , β = n + and ϕ = ω , (16) then the equation has a Liouvillian first integral given byI = | V | − n (cid:183) exp (cid:181) n − ϕ u n nV (cid:182) V + ( n − u F (cid:181) n , 1 + n , 2( n − ϕ u n nV (cid:182)(cid:184) , (17) where V = u + β ˙ u + ϕ u n . The above integral is real, and its only singularity corre-sponds to the invariant curve V = . The appearance of hypergeometric functions is quite remarkable – it shows howrestrictive the notion of analytic, meromorphic or algebraic integrability can be.Although various parametric forms of solutions of polynomial dynamical systemscan be given in terms of hypergeometric functions [11], this seems to be the firstexample of a natural (i.e. not artificially contrived) system with first integrals ofsuch form. This is the price we have to pay for the transition from relatively simple,but time-dependent, integrals to time-independent ones, which permit us to call anautonomous system integrable in the standard sense.A starting point in integrability analysis might be looking in the class of analyticfunctions, but because trajectories converge to the node at the origin, this approachwill not work here. As the systems are of dimension two, it is not possible to directlyapply the basic methods of differential Galois theory [12] either, because the firstnormal variational equation will be of order 1 and thus soluble. Higher variationalequations could be used, but in addition to complicating the analysis, it will still belimited to meromorphic first integrals. To make progress, we will have to turn tothe Darboux polynomials, which are most often used in the context of rational firstintegrals, but can lead to algebraic or even Liouvillian expressions.4he remainder of the article is devoted to introducing the basic machinery (Section2.) and providing a general criterion for the appearance of hypergeometric firstintegrals in polynomial systems (Lemma 1). The proofs of the theorems are given inSections 2 through 4, and the central criterion is further extended in Lemma 2. Thepossibility of reduction to more elementary functions is discussed in the final section. The first step in the construction of first integrals is to consider the dynamical systemcorresponding to (1) ˙ x = x y = : P ,˙ y = − ω − y (1 + y ) − x = : Q , (18)where x = u and y = ˙ u / u . The vector field associated to the flow defines a derivationover the polynomial ring (cid:67) [ x , y ], and one can look for the Darboux polynomials of thederivation D : = P ∂ x + Q ∂ y . If enough of them are found, the second step is to combinethem into a first integral.To briefly review the general setting, consider a d -dimensional system or, equiva-lently, a derivation over complex polynomials of d variables. A Darboux polynomialF is defined as an elements of (cid:67) [ x , . . . , x d ], such that DF = K F , for some K ∈ (cid:67) [ x , . . . , x d ], (19)and K is called the cofactor . Since along a solution one has DF = ˙ F , it follows fromthe definition that F = K ≡ F is just a polynomialfirst integral. Another basic property holds for the product: D ( F F ) = ( K + K ) F F ,so that F F is itself a Darboux polynomial. In particular, any integral power F m isa Darboux polynomial with the cofactor mK . A converse result can also be proven: if F is reducible, its factors must be Darboux polynomials too.Importantly, the existence of such polynomials allows one to analyze integrabilitythrough rational functions, because if f / g ∈ (cid:67) ( x , . . . , x d ) is a first integral, thennecessarily f and g must be Darboux polynomials with the same cofactor; conversely,if there are enough Darboux polynomials, so that K ’s are linearly dependent over (cid:90) ,a rational first integral exists. For the full Darboux theorem see [13].Going beyond polynomials, F q , with rational q , is an algebraic function which stillhas a polynomial cofactor. Indeed for any function f , Liouvillian over (cid:67) ( x , . . . , x d ),there might exist a polynomial K such that D f = K f ; for brevity the term
Darbouxpair will be used to refer to { f , K } then. 5he last general remark to be made is that the degree of K has to be lower thanthat of the derivation, and finding variables for which D is quadratic will simplifythings considerably. However, neither this, nor d = F , the equation DF = K F can be solved term by term. It is a straightforward calculation to verify thatwhen ω = the following quadratic Darboux polynomials exist F = x , K = y , F = x + + y ) , K = − − y . (20)As mentioned, the irreducible factors of F are themselves Darboux polynomials F = ± (cid:112) x + y , K = − ± (cid:112) x − y , (21)but with a view to finding a real first integral, F will be used.It should also be noted, that there is no effective general tool to give the bounds onthe degree of F , so when ω (cid:54)= the question of existence of higher degree F remainsopen. In general it is a difficult task to exclude all possible degrees, see for example[15], but fortunately the goal here is not to find all the polynomials or rational firstintegrals.Thanks to the product property, Darboux polynomials (20) can be combined, andthe new cofactor can be made constant D (cid:161) x (9 x + + y ) ) (cid:162) x (9 x + + y ) ) = −
43 , (22)so that we immediately have J : = x (9 x + + y ) ) = e − t /3 J , (23)where J = J ( x (0), y (0)) is equivalent to I D of (6).If K ’s are dependent over (cid:90) , the resulting cofactor can even be made zero, so arational first integral is found (or at least algebraic, were they dependent over (cid:82) ).This is not the case here, and there are no new Darboux polynomials of degree 3either, that is, they are all products of F and F . In principle, one could try lookingat higher degrees, but as it turns out this will not be necessary.6n important thing to notice is also that the following combination of (20) isrelated to the divergence of the flow (18) D ( F / F ) = − ( ∂ x P + ∂ y Q )( F / F ). (24)This means that a Liouvillian first integral can be obtained, as described by Theorem1 of [13] with F / F being the integrating factor of the form P d y − Q d x . Instead ofapplying that theorem, however, a slightly different derivation will be given here,which leads directly to a more concise expression for the first integral. It relies on aresult valid in a very general setting: Lemma 1.
Let D = (cid:80) di = P i ∂ i be a derivation which admits two Darboux polynomialsf and f with cofactors k and k , respectively. If f : = f − f is also a Darbouxpolynomial with cofactor k , such that α k + α k + α k = α ∈ (cid:67) , α i ∈ (cid:67) not all zero , (25) and f is the cofactor of f / f , i.e., k − k = f , (26) then the dynamical system ˙ x = P i ( x ) associated with the derivation D has a Liouvillianfirst integral, expressible locally in terms of the hypergeometric function.Proof. The first condition means that we have a function J = (cid:81) f α i i such that D J = α J or J ( t ) = e α t J (0). Let us thus take the time integral of J and re-express itwith f i and a new variable ζ : = f / f to get (cid:90) J d t = (cid:90) f α f α f α d t , J α + C = (cid:90) f α f α f α d ζ ( f − f ) ζ = (cid:90) ( f / f ) α f α + α − f α d ζ (1 − ζ ) ζ , J α + C = (cid:90) ζ α − (1 − ζ ) α − f α + α + α − d ζ , (27)where the determination of the complex argument of ζ or 1 − ζ was ignored, with theunderstanding that the form of the integral remains the same save for a multiplicativeconstant. This constant depends on the region in which ζ (and hence x ) lies, which iswhy the word “locally” is necessary in the conclusion.Because α i are defined up to rescaling, we can take α + α + α = I = J α − ζ α α F (1 − α , α , α + ζ ), (28)7s the sought first integral. Because the Euler representation is obtained from anindefinite integral, the result is a Liouvillian function of ζ , which in turn is a rationalfunction of the original variables.Thus, in the standard case of equation (1), the above lemma can be applied with f = − F , f = − F , f = (cid:112) F , (29)but the resulting first integral will then explicitly contain the imaginary unit. Aslight modification is required if one insists on real expressions, and it can be effectedfor the more general equation (2). Proof of Theorem 1.
Let us introduce a new set of variables x = u n − and y = ˙ u / u ,in which system (2) reads ˙ x = ( n − yx ,˙ y = − ω − y (1 + y ) − x . (30)As before, F = x is a Darboux polynomial, and direct computation shows that F = x + ω (2 + ( n + y ) (31)is another one if the condition ω = n + n + is satisfied. As above, a function J with constant logarithmic derivative can be found to be J = F n − F , with D J = − ( n + ω J . (32)The new variable to use is ζ = x x + ω (2 + ( n + y ) , (33)for which we have ˙ ζ = D ζ = n + n + + ( n + y ) ζ , (34)and additionally ω (2 + ( n + y ) = F − F = F (1 − ζ ). (35)This makes it possible to calculate the time integral of J α , with α to be determined, (cid:90) J α d t = (cid:90) F α n − F α d t = (cid:90) (cid:181) F F (cid:182) α n − F αγ d ζ ˙ ζ , C − J α ( n + ω α = ς n + n + α − n ω αγ (cid:90) ζ α n − − (1 − ζ ) − αγ (2 + ( n + y ) αγ − d ζ , (36)8here γ = ( n + n − | ς | = u and ˙ u .Taking α = γ ) in the above leads again to the Euler integral and the hypergeo-metric functions C = n + n − J α − ς n + n + n − n + ω (1 − ζ ) F (cid:181)
12 , nn + − ζ (cid:182) , or (cid:101) C = n + n − J α + ς ( n +
3) 4 − n + ω ζ n + F (cid:181)
12 , 1 n + n + n + ζ (cid:182) . (37)These are two forms of the same first integral, but the former is more convenientaround ζ =
1, which corresponds to the line 2 + ( n + y =
0, and has a singularity at ζ =
0; conversely, the latter works around ζ = x =
0) and has a singularity at ζ = z and J in terms of x and y and then u and ˙ u , we againignore the complex arguments and expand the powers freely. For (cid:101) C ( C is analogous)this gives I = n + n − V − n + (cid:183) V + ς ( n − ω u F (cid:181)
12 , 1 n + n + n + u n + V (cid:182)(cid:184) , (38)with V = u n + + ω (2 u + ( n +
3) ˙ u ) . Instead of dealing with all the regions of ζ or x and y to determine ς it seems easier to differentiate I and ensure that thepseudo-polynomial expression in u and ˙ u vanishes identically. This happens for ς = u + ( n +
3) ˙ u (cid:112) V (cid:181) (2 u + ( n +
3) ˙ u ) V (cid:182) − . (39)When n is odd, V is always positive, so this reduces to ς = sign(2 u + ( n +
3) ˙ u ); when n is even, V = C , which lies in the left half-plane andhas a cusp at the origin. V is positive to the right of this curve, so (38) applies therewith the same sign factor. For C , and consequently I in (9), the factor becomes ς (cid:48) = u (cid:181) u n + V (cid:182) − n + V − n + , (40)which is just sign( u ) for odd n ; when n is even, I is to be used to the left of C , whereboth u and V are negative, so ς (cid:48) simplifies to ( − n /( n + as in the theorem. Bothhypergeometric functions are real when their argument is less than 1, so only in thelast case does the integral acquire a (constant) complex phase via the roots of V .9n interesting difference between between this and Lemma 1 is that we haveonly used a relation between K − K and F − F , without assuming that the latteris linked with F . Because J involved only F and F that was enough to express theintegrand as a function of ζ only. This observation will be crucial in proving a moregeneral lemma in section 4.It should also be noticed that the Gaussian integral happens to be expressible viathe incomplete beta function, namely (cid:90) z ζ n + − (1 − ζ ) − d ζ = B z (cid:161) n + , (cid:162) . (41)Together with the special case n = n is shown in Figure 1. As the origin is an attracting node, there cannot exist aglobal analytic first integral. Upon passing between the half-planes, the values ofthe two integrals I and I need to be adjusted if one wants to keep the formulaeintact, alternatively they can be regarded as multivalued functions. The slightlymore complicated situation with even n is shown in Figure 2, in addition to the lineof discontinuity there appears an invariant curve with another critical point on it.Finally, one has to remember that to generalize the u term in (1) to arbitraryexponents, while keeping the “harmonic” character of the force, one should take | u | n − u instead of just u n . The argument of the hypergeometric function, whichcorresponds to ζ , then satisfies 0 ≤ | u | n + / V ≤ All of the above can be immediately applied also to the Duffing-van der Pol equation(10), although just the existence of a time-dependent first integral will not be enough.As in [8], the exponents will have to satisfy m = n −
1, so that excludes the classicalvan der Pol ( m = n = m = n =
3) oscillators.
Proof of Theorem 2.
Adopting the same variables as before, i.e, x = u n − and y = ˙ u / u , the system is now ˙ x = ( n − yx ,˙ y = − ω − y (1 + y ) − x (1 + β y ). (42)10igure 1: The phase space of the Duffing oscillator with n = u and ˙ u coordinates.The left diagram shows level sets of I , the right one those of I , as defined in Theorem 1.Notice the mismatch between gradient directions. The Darboux polynomial F = x is self-evident, and if the condition ω = n ( β − n ) β − , (43)holds, a second linear one can be found by direct computation: F = β x + n β (1 + y ) − n , K = − ( n + β y ) β − . (44)These two are enough for the construction of (11), because we have D ( xF n − ) xF n − = n (1 − n ) n + F / F ,or a similar combination, is a function of y alone so not necessarily a Darbouxpolynomial. At the same time, F − β F , which could be the candidate for f − f , islinear in y but in order for it to be a Darboux polynomial, an additional condition isnecessary: ω = ( β − β − . (46)It is independent of the one in (43), and it guarantees the existence of the Darbouxpair F = + β y , K = β − − − y − β x . (47)11igure 2: The phase space of the Duffing oscillator with n = u and ˙ u coordinates; I and I of Theorem 1 are used in the right and left diagrams, respectively. The discontinuityline is drawn in blue to distinguish it from the invariant curve in red. The additional criticalpoint is depicted by a red square. Notice that just F and F are insufficient for the time-dependent integral underconsideration, because no linear combination of their cofactors can be made constant.It should also be said that unless a full characterization of the system’s Darbouxpolynomials is given, it remains an open question if the condition (43) alone is notenough to proceed with the proof.If both (43) and (46) are to be satisfied, it follows that β = n + ω is strictly determined by the degree n .Is is now straightforward to take f = − ( n + F , f = nF /( n + f = F /( n + ζ = f / f and J = f n f − n , which leadsto the following Euler integrals n + − n J = ( n + − n (cid:90) ζ n − (1 − ζ ) − n d ζ = ς n ( n + n − (1 − ζ ) − n ζ n F (cid:161)
1, 1, 1 + n , ζ (cid:162) + C = ς (cid:48) n ( n + − n n − ζ n (1 − ζ ) − n F (cid:161)
1, 1, 2 − n , 1 − ζ (cid:162) + (cid:101) C . (49)12 difference with the previous situation is that | ζ | might be greater than 1, even whenwe take x = | u | n − . Still, the above hypergeometric functions can be continued alongthe real axis past ζ = − ζ = ζ = ∞ .The former corresponds to f = f ⇐⇒ F =
0, and the latter to f = ⇐⇒ F =
0, sothey are both invariant sets. In the original variables, they become ω ( u + β ˙ u ) + u n = β ˙ u + u =
0, respectively. These curves separate the phase space into regions inwhich one of the above hypergeometric function can be chosen. Finally, substitutingfor ζ , x and y , the sign factors can be determined as in the previous proof, and turnout not to depend on the region giving the stated formulae. Lastly, the integralsmight acquire a complex phase due to fractional powers of V = u n + ω ( u + β ˙ u ), butsince this is just a multiplicative constant, its absolute value can be taken on eachside of V = (cid:90) z ζ n − (1 − ζ ) − n d ζ = B z (cid:161) n , 1 − n (cid:162) , (50)although it is more convenient to use the Gauss function, so that the fractional powerof V can be written as a common factor. In fact, this beta function turns out to beelementary, albeit with no simple general formula, as is shown in Section 5.Like before, for non-integer n , or to keep the sign as in the harmonic case, u n canbe replaced by | u | n − u . A plot of the level sets of I and I is presented in Figure3. Both first integrals are valid everywhere between the red curves, and the twoformulae merely reflect the fact that they are not always real-valued. As opposed tothe previous system, there is no need for the regions of definition to overlap, becausethe red curves, being invariant sets, are impassable barriers. The third system presents a new challenge, because of non-polynomial Darbouxelements. To wit, in the coordinates x = u n − , and y = ˙ u / u it reads˙ x = ( n − yx ,˙ y = − ω − y (1 + y ) − x (1 + β y ) − ϕ x . (51)If, in addition to both the previous conditions β = n +
1, and ω = n ( n + − , the newparameter satisfies 4 ϕ = ω − , (52)13igure 3: The phase space of the Duffing-van der Pol oscillator with n = u and ˙ u coordinates. The left diagram shows level sets of I as shaded, the right one those of I , bothdefined in Theorem 2. then, there exist three Darboux pairs: F = x , K = ( n − y , F = + β y + ϕ x , K = ( β − − − y − ( β /2) x , F = exp (cid:161) ϕ F / F (cid:162) , K = ( β /2) x . (53)Moreover, they generate a time-dependent first integral through J = F α F α F α , with D J = n (1 − n )1 + n J , (54)where the exponents were chosen to be( α , α , α ) = (1, n − n − k − k had to bereexpressed as a function of f i ; likewise, the time-dependent integral contained f f / f , J or ( f − f ) α , and they can be grouped as we please, with theaim of integration in the variable ζ . This suggests the following generalization: Lemma 2.
Let D = (cid:80) di = P i ∂ i be a derivation which admits at least two Darbouxelements which can be combined to yield Darboux pairs { f , k } and { f , k } such that: the element J : = f f has a constant cofactor α ; the cofactor of ζ : = f / f satisfiesk − k = f γ L ( ζ ), (cid:67) (cid:51) γ (cid:54)=
0, (56) for some Liouvillian L. Then, the dynamical system ˙ x = P i ( x ) associated with thederivation D has a Liouvillian first integral.In particular, a binomial L ( ζ ) = ζ a (1 − ζ ) b leads to the Gauss function, whileL ( ζ ) = ζ a e b ζ to the Kummer function in the first integral.Proof. The element J considered as a function of time satisfies J ( t ) = e α t J (0), sothe integral of J γ can be formally transformed, using ζ = f / f as follows (cid:90) J γ d t = (cid:90) f γ f γ ( k − k ) ζ d ζ = (cid:90) ( f / f ) γ f γ f γ ζ L ( ζ ) d ζ J γ γα − C = (cid:90) ζ γ − L ( ζ ) − d ζ . (57)The same considerations of complex arguments as in Lemma 1 apply, so a suitableconstant ς of modulus 1 will have to be added once a particular region of ζ is fixed.The two special cases then give the time-independent first integral I through J γ ςγα − I = (cid:90) ζ γ − a − (1 − ζ ) − b d ζ = B ζ (cid:161) γ − a , 1 − b (cid:162) = ζ γ − a γ − a F ( γ − a , b , 1 + γ − a , ζ ), (58)or J γ ςγα − I = (cid:90) ζ γ − a − e − b ζ d ζ = − b a − γ Γ ( γ − a , b ζ ) = ζ γ − a γ − a F (cid:161) γ − a , γ − a + − b ζ (cid:162) , (59)where Γ ( s , z ) is the incomplete gamma function.15oth the previous proofs are special cases, and they are important prelimi-nary results showing how to combine the Darboux polynomials. In Lemma 1,denoting the “old” polynomials by g i , we can take f = g ( α + g ( α − g α /23 , and f = g ( α − g ( α + g α /23 , so that ζ = f / f = g / g , and J = f f = g α g α g α , whilethe relation between the cofactors yields f ζ − α (1 − ζ ) α + α = g α + α + α = g − g = k − k . (60)when α + α + α = f = F F ( n − , f = F ( n + , and then k − k = K − K = (cid:113) n + ( F − F ) = (cid:113) n + f n + (cid:112) − ζ . (61)The crucial quantity is always k − k , which is polynomial in the dynamical variables,but not necessarily a polynomial, or even algebraic, in f i . This is ostensibly so in thepresent system (51), to which we now turn. Proof of Theorem 3.
The exponential factor suggests the choice of ζ = F / F , andLemma 2 can be used with f = F F n − F n − , f = F n F n − . (62)The relation between the cofactors is k − k = K − K = f n exp (cid:181) − n ) ϕ n ζ (cid:182) , (63)so, by Lemma 2, the first integral is C = n + − n J n − ς ( n + ζ n F (cid:181) n , 1 + n , 2( n − ϕ n ζ (cid:182) . (64)or, going back to the original variables, I = V − n (cid:183) exp (cid:181) n − ϕ u n nV (cid:182) V + ς ( n − u F (cid:181) n , 1 + n , 2( n − ϕ u n nV (cid:182)(cid:184) , (65)where V = u + β ˙ u + ϕ u n . The curve given by V = V − n factor. Differentiating then gives ς = n = n = n , or just to keep a single critical point atthe origin, a change of u n to | u | n − u is necessary. The invariant curve corresponds tothe irregular singularity of the Kummer function at infinity, so a series expansionaround it is not readily available due to the Stokes phenomenon. Still, as with theexponential function, the series around zero has infinite radius of convergence.16igure 4: The phase space of the generalized Duffing-van der Pol oscillator for n = n = u and ˙ u coordinates. All contours are given by I of Theorem 3, thered curve is the invariant curve at the integral’s singularity, and the additional critical pointis indicated with a square. Having obtained the hypergeometric expressions, the natural question to ask iswhether they can be reduced to elementary functions. In the case of F , the answerfollows from a result of Chebyshev’s [17]: Theorem 4.
If p, q and r are rational numbers and a and b are nonzero realnumbers, the indefinite integral (cid:82) z p ( a + bz r ) q d z is elementary if and only if at leastone of ( p + r, q, or q + ( p + r is an integer. In the first system, a = − b = r = p = − + n +
1) and q = − n + − − + n +
1) – none of them areintegers when n >
1, so the hypergeometric function is truly transcendental in thiscase.In the second system, b = − a = r = p = − + n and q = − n , which meansthat q + ( p + r = n . However, the theorem still requires that theexponents be rational, and the specific change of variables that make the integrandrational depends on those exponents. In the central case of integer n > ζ = + z n ), and the relevant expression is I z = (cid:90) ζ n − (1 − ζ ) − n d ζ = − (cid:90) nz n − z n + z . (66)17he evaluation of such an integral is straightforward for a specific value of n , e.g. n = I z = − (cid:179)(cid:112) − ζ (cid:180) , but it cannot be given as a simple expression forsymbolic n – other than by using the hypergeometric or beta functions.Additionally, the transcendental functions of the first theorem can become inversesof elliptic functions. The three cases when this happens for B ζ (cid:161) n + , (cid:162) are given in[11], and correspond to n =
2, 3, 5. This is best seen on the initial case, which admitsa change to the Hamiltonian system (5), and can be integrated for energy E by Z = (cid:90) (cid:112) W (cid:112) E − W = B ζ (cid:161) , (cid:162) E , (67)where, in the transformed variables, ζ = W /(4 E ). This is an inversion of the directsolution of the Hamiltonian system in terms of the Jacobian elliptic function W =(cid:112) E sn( E Z ).Finally, the Kummer function of Theorem 3 is transcendental for integer n > Γ (cid:161) n , ζ (cid:162) = n (cid:90) ∞ s e − s n d s , (68)where ζ = s n . The Gauss error function is a familiar example of the above for n = Acknowledgements
This work was supported partially by the grant No. DEC-2013/09/B/ST1/04130 ofthe National Science Centre of Poland, and partially by the Japan Society for the Pro-motion of Science, Grant-in-Aid for Scientific Research (B) (Subject No. 17H02859).
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