We study the local analytic integrability for real Li\'{e}nard systems, $\dot x=y-F(x),$ $\dot y= x$, with $F(0)=0$ but $F'(0)\ne0,$ which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the $[p:-q]$ resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the $[p:-q]$ resonant saddle into a strong saddle.The first author is partially supported by a MINECO/FEDER grant number MTM2014-
53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR-1204. The
second author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a
MINEC0 grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568

Giné, Jaume

Llibre, Jaume

Applied Mathematics Letters

English

We study the local analytic integrability for real Li\'enard systems, x=y-F(x), y= x, with F(0)=0 but F'(0)0, which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the [p:-q] resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the [p:-q] resonant saddle into a strong saddle

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ON THE INTEGRABILITY OF LIE´NARD SYSTEMSWITH A STRONG SADDLEJAUME GINE´1 AND JAUME LLIBRE2Abstract. We study the local analytic integrability for real Lie´nard systems, x˙ = y−F (x),y˙ = x, with F (0) = 0 but F ′(0) ̸= 0, which implies that it has a strong saddle at theorigin. First we prove that this problem is equivalent to study the local analytic integrabilityof the [p : −q] resonant saddles. This result implies that the local analytic integrability ofa strong saddle is a hard problem and only partial results can be obtained. Neverthelessthis equivalence gives a new method to compute the so-called resonant saddle quantitiestransforming the [p : −q] resonant saddle into a strong saddle.1. Introduction and main resultsTwo of the main problems about the planar Lie´nard diﬀerential systems are to knowwhether they are integrable or not, and to give criteria for controlling their number of limitcycles. In this work we deal with the ﬁrst problem. For more information on the secondmentioned problem, see [1]. Lie´nard diﬀerential systems appeared in electrical circuits withnonlinear elements, but later many other situations have been modeled by these type ofdiﬀerential equations, see [7] and references therein.Assume that a real planar analytic diﬀerential system has a weak focus. It is well-knownthat this singular point is a center if and only if the equation has an analytic ﬁrst integraldeﬁned in a neighborhood of this point, see for instance [18]. Consequently the center problemfor non-degenerate singular points is equivalent to the local analytic integrable problem forsuch singular points.The following theorem characterizes the centers for the classical real analytic Lie´nard sys-tems, see the proof in [3, 5, 6].Theorem 1 (Cherkas). Consider the diﬀerential systemx˙ = y + F (x), y˙ = −x,with F (x) analytic at zero and F (0) = F ′(0) = 0, then the origin is a center if and only if Fis an even function.Recently have been proved a similar result, but for classical Lie´nard systems with a weaksaddle at the origin, that is, systems of the formx˙ = y + F (x), y˙ = x,with F analytic and F (0) = F ′(0) = 0. In this case the eigenvalues are ±1. Recall that asaddle point is weak if its eigenvalues are ±λ with 0 ̸= λ ∈ R. The duality between centersand saddles transforms one center into one saddle when we consider the system in C2, see [7].2010 Mathematics Subject Classiﬁcation. Primary 37C15. Secondary 34A34; .Key words and phrases. Center problem, analytic integrability, strong saddle, Lie´nard equation, resonantsaddle.1This is a preprint of: “On the Integrability of Lie´nard systems with a strong saddle”, Jaume Gine´,Jaume Llibre, Appl. Math. Lett., vol. 70, 39–45, 2017.DOI: [10.1016/j.aml.2017.03.004]2 J. GINE´ AND J. LLIBREThe result obtained in [7] is the following:Theorem 2. Consider the analytic diﬀerential systems in C2 of the form(1) x˙ = y + F (x), y˙ = ax,with 0 ̸= a ∈ C and where F (x) is an analytic function without linear and constant terms.System (1) is locally integrable at the origin if and only if F (x) is an even function of x.The result covers the classical Lie´nard case in R2 with a weak focus at the origin whichcorresponds to a = −1, and the weak saddle case that has a = 1, and also extends to C2 someof the results obtained in [16].In [7] Theorem 2 is generalized to get the characterization of the real analytic integrableweak saddles for general Lie´nard systems in R2, that is, for systems of the form(2) x˙ = y + F (x), y˙ = g(x),where F and g are analytic functions of x, F (x) without linear and constant terms and withg(0) = 0 and g′(0) ̸= 0. This generalization use the ideas given in [3, 6], see also [2, 8, 9], andis also recently proved in [16]. The result can be written in the following way.Corollary 3. System (2) has an integrable resonant weak saddle (resp. weak focus) at theorigin if and only if g′(0) > 0 (resp. g′(0) < 0) and F (x) = ϕ(G(x)), for some analyticfunction ϕ with ϕ(0) = 0 and where G(x) =∫ x0g(ξ)dξ.Moreover in [7], using Lu¨roth theorem as in [4], an eﬀective characterization of real in-tegrable polynomial Lie´nard systems with a weak saddle is obtained. In [10] some otherintegrable Lie´nard-like complex systems with a weak saddle are also characterized.In this work we aim to characterize the analytic integrability of the real diﬀerential systemsof the form(3) x˙ = y + F (x), y˙ = x,where F (x) is an analytic function of x with F (0) = 0 but F ′(0) ̸= 0 which implies that system(3) has a strong saddle at the origin. Without loss of generality we can take F ′(0) = 1/c− c,with c ∈ R+ \ {1}. We avoid the case c = 1 which corresponds to the weak saddle case. Thenthe eigenvalues λ1 and λ2 of the linear part are −c < 0 < 1/c. It is well-known, see [13, 17],that if −λ1/λ2 ̸∈ Q+ then the diﬀerential equation (3) has no local analytic ﬁrst integral in aneighborhood of the origin. Consequently a necessary condition for having an analytic ﬁrstintegral at the origin is c2 = q/p ∈ Q+. In fact, if F (x) = (√p/q −√q/p) x it is not diﬃcultto prove thatH(x, y) =(√p x+√q y)q(√p y −√q x)pis a ﬁrst integral of the linear Lie´nard system.Before stating our main result, we introduce some notation and deﬁnitions. The centerproblem for analytic diﬀerential systems in R2 with a nondegenerate singular point, that is,systems of the form x˙ = −y + · · · , y˙ = x+ · · · , where the dots mean higher order terms, canbe transformed by the complex change of variable u = x + iy, v = x − iy into the complexdiﬀerential system of the form u˙ = u + · · · , v˙ = −v + · · · . The natural generalization of theabove system is to consider analytic diﬀerential system in C2 of the form(4) u˙ = λu+ · · · , v˙ = −µ v + · · · ,where λ, µ ∈ C \ {0}. As we have said a necessary condition to have analytic integrality atthe origin of system (4) is λ/µ = p/q ∈ Q+ with gcd(p, q) = 1. This case is called [p : −q]INTEGRABILITY OF LIE´NARD SYSTEMS WITH A STRONG SADDLE 3resonant case. In this situation the local analytic integrability is sometimes possible withsome more necessary conditions. We describe how to obtain these necessary conditions.The [p : −q] resonant case can be written as(5) u˙ = p u+ · · · , v˙ = −q v + · · · ,after a scaling of time if necessary, with p, q ∈ Z+. The linear part of system (5) has theanalytic ﬁrst integral H0 = uqvp and the inverse integrating factor V0 = u1−qv1−p. Now wecan search conditions for the existence of an analytic ﬁrst integral H(u, v) = H0 + · · · forsystem (5) and we get the equationH˙ = v1H20 + v3H30 + · · · ,where vi are the so-called [p : −q] resonant saddle quantities which are polynomials in thecoeﬃcients of system (5). Therefore we have a formal analytic resonant saddle if all the viare zero, see [12, 19] and references therein. Moreover in this case we have also the existenceof a local analytic ﬁrst integral, see [18].The ﬁrst result of this work is the following.Theorem 4. System (3) with a strong saddle at the origin, that is, with F (0) = 0 butF ′(0) = 1/c− c ̸= 0 and with c2 = q/p ∈ Q+ can be transformed into a system with a [p : −q]resonant saddle at the origin.Theorem 4 is proved in section 4.The analytic integrability problem of a [p : −q] resonant saddle is a very hard problem asthe several papers dedicated to this problem in the last years have shown, see for instance[12, 14, 15, 19, 20] and references therein. Hence the ﬁrst consequence of Theorem 4 is thatthe analytic integrability of a strong saddle is as diﬃcult as the analytic integrability problemof the [p : −q] resonant saddle. The second consequence is that we can derive a new methodto compute the resonant saddle quantities of a system with a [p : −q] resonant saddle as weshow in section 2.In [7] it was proposed the following open problem: Are there nonlinear integrable systemsinside the family of system (3)? The following proposition establishes that in a particularcase the unique integrable case is the linear one.Proposition 5. The Lie´nard diﬀerential system with a strong saddle at the originx˙ = y + (1/c− c)x+ a2x2 + a3x3, y˙ = x,with c2 = q/p = 2/1 ∈ Q+ has a unique integrable case which corresponds to a2 = a3 = 0.Moreover, for several [p : −q] resonant saddles of homogeneous and non homogeneousdiﬀerential systems that can be transformed into a system of the form (3) it has been checkedthat the only integrable case is when F (x) = ax. Hence we establish the following conjecture.Conjecture 6. The unique integrable case of the Lie´nard system (3) is the linear one.What is not true is that the unique integrable case of a general strong saddle be the linearone. Indeed, if we consider more general systems of the form(6) x˙ = y + F (x, y), y˙ = x,where F (x, y) = ax+· · · with a ̸= 0, here the dots mean higher order terms which implies thatsystem (6) has also a strong saddle at the origin, these systems can have analytic integrablesaddle diﬀerent from the linear one as the following example shows.4 J. GINE´ AND J. LLIBREProposition 7. The diﬀerential system(7) x˙ = y − x√2+bx2√2+ 2bxy +√2by2, y˙ = x,has an analytic integrable strong saddle at the origin because has the analytic ﬁrst integralH = e12b3(4√2+6by+3(√2+2by)(1+b(x+√2y))2− 12(√2+by1+b(x+√2y))(1 + b(x+√2y))−3√2b3 .It is easy to check that the function H in Proposition 7 is a ﬁrst integral of system (7).2. A new method to compute the resonant saddle quantitiesWe consider the following analytic diﬀerential system with a [p : −q] resonant saddle at theorigin(8) u˙ = p u+∞∑i=2aijuivj, v˙ = −q v +∞∑i=2bijuivj,where p, q ∈ Z+ and aij, bij ∈ R. Now we deﬁne c2 = q/p and we made the scaling of timet→ cp t and system (8) takes the form(9) u˙ =uc+∞∑i=2a˜ijuivj, v˙ = −c v +∞∑i=2b˜ijuivj,where a˜ij, b˜ij ∈ R. Next we do the linear change of variablesx = −−u+ cv1 + c2, y =cu+ v1 + c2,and system (9) becomes(10) x˙ = y +(1c− c)x+∞∑i=2cijxiyj, y˙ = x+∞∑i=2dijxiyj.Then we implement the change of variablesU =x√2− y√2, V =x√2+y√2,and system (10) becomes(11)U˙ =1− 2c− c22cU +1− c22cV +∞∑i=2c˜ijUiV j,V˙ =1− c22cU +1 + 2c− c22cV +∞∑i=2d˜ijUiV j,where c˜ij, d˜ij ∈ R. Next we introduce the complex change of variables U = X+iY , V = X−iYand system (11) can be written as(12) X˙ = −Y + i(c− 1c)X +∞∑i=2˜˜cijXiY j, Y˙ = X +∞∑i=2˜˜dijXiY j,where here ˜˜cij,˜˜dij ∈ C. A ﬁrst integral of the linear part of system (12) is given byH0 =(√qpX + iY)p(X − i√qpY)q,INTEGRABILITY OF LIE´NARD SYSTEMS WITH A STRONG SADDLE 5and consequently a ﬁrst integral of system (12) must be of the form H = H0 + · · · where thedots are higher order terms, see [11]. Notice that system (12) has complex parameters andit has the linear part of a center-focus type. Hence the classical method of passing to polarcoordinates and to ﬁnd obstructions to have a formal ﬁrst integral is now possible.Therefore in system (12) we take the polar coordinates X = r cos θ and Y = r sin θ and wepropose the Poincare´ power series(13) H(r, θ) =∞∑m=2Hm(θ)rm,where H2(θ) = 1/2 and Hm(θ) are homogeneous trigonometric polynomials respect to θ ofdegree m. Computing the derivative of H we haveH˙(r, θ) =∞∑k=2V2kr2k,where the V2k’s are the obstructions to have H as a formal ﬁrst integral of system (12). ThisV2k are the saddle quantities which are polynomials in the parameters of system (12), see [18].However these quantities correspond to the saddle quantities of the original system (8) andconsequently are polynomials in the real parameters of system (8).The equivalence that establishes Theorem 4 also helps to understand why it is more diﬃcultto classify of the integrable systems of a [p : −q] resonance when we increase p or q. Whathappens is that the ﬁrst integral of the linear part increases also its degree, and hence thesaddle quantities appear for degrees higher in the terms of the proposed power series (13).Consequently the saddle quantities are polynomials of higher degree in the variables of system(8), and it is more diﬃcult to ﬁnd the zeros of the ideal generated by these saddle quantities.Proof of Proposition 5. Using the method described in the previous section we ﬁnd that theﬁrst two nonzero saddle quantities areV6 =√2a22 + 21a3,V8 = 177017√2a62 + 4078901a42a3 + 3855267√2a22a23 − 33480a33,modulo the previous one zero, which implies a2 = a3 = 0. 3. Proof of Theorem 4The proof is based on apply several transformations to system (3). First we do a rotationof angle φ = π/4 which is given by the linear changeu =x√2− y√2, v =x√2+y√2,and system (3) with F (0) = 0 but F ′(0) = 1/c− c ̸= 0 takes the form(14) u˙ =12(−2 + 1c− c)u+ 1− c22cv + · · · , v˙ = 1− c22cu+12(2 +1c− c) v + · · · .The eigenvalues of the linear part are λ1 = 1/c, and λ2 = −c whose eigenvectors arev1 =(1− c1 + c, 1), v2 =(c+ 1c− 1 , 1),6 J. GINE´ AND J. LLIBRErespectively. Now we propose the linear change of variablesX =1− c1 + cu+ v, Y =c+ 1c− 1u+ v,and the scaling of time t→ ct, and system (14) takes the form(15) X˙ = X + · · · , Y˙ = −c2 Y + · · · .Taking into account that c2 = q/p ∈ Q+ doing a second scaling of time we obtain the system(16) X˙ = pX + · · · , Y˙ = −q Y + · · · .which corresponds to a [p : −q] resonant saddle and the proof of the theorem is completed.4. AcknowledgementsThe ﬁrst author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR-1204. Thesecond author is partially supported by a MINECO grants number MTM2016-77278-P andMTM2013-40998-P, an AGAUR grant number 2014SGR-568, and the grant FP7-PEOPLE-2012-IRSES 318999.References[1] M. Caubergh, Hilbert’s sixteenth problem for polynomial Linard equations, Qual. Theory Dyn. Syst. 11(2012), no. 1, 3–18.[2] L.A. Cherkas, On the conditions for a center for certain equations of the form yy′ = P (x) +Q(x)y +R(x)y2, Diﬀer. Uravn. 8 (1972), 1435-1439; Diﬀer. Equ. 8 (1972), 1104–1107.[3] L.A. Cherkas, Conditions for a Lie´nard equation to have a centre, Diﬀer. Equ. 12 (1976), 201–206.[4] C.J. Christopher, An algebraic approach to the classiﬁcation of centres in polynomial Lie´nard systems,J. Math. Anal. Appl. 229 (1999), 319–329.[5] C. Christopher, N.G. Lloyd, Small-amplitude limit cycles in polynomial Lie´nard systems, NoDEA 3(1996), 183–190.[6] C.J. Christopher, N.G. Lloyd, J.M. Pearson, On a Cherkass method for centre conditions, Non-linear World 2 (1995), 459–469.[7] A. 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Zoladek, The problem of center for resonant singular points of polynomial vector ﬁelds, J. DiﬀerentialEquations 137 (1997), no. 1, 94–118.1 Departament de Matema`tica, Universitat de Lleida, Avda. Jaume II, 69; 25001 Lleida,Catalonia, SpainE-mail address: gine@matematica.udl.cat2 Departament de Matema`tiques, Universitat Auto`noma de Barcelona, 08193 Bellaterra,Barcelona, Catalonia, SpainE-mail address: jllibre@mat.uab.cat

On the Integrability of Liénard systems with a strong saddle

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