Recently A.R.Chouikha gave a new characterization of isochronicity of center
at the origin for the equation $x"+g(x)=0$, where $g$ is a real smooth function
defined in some neighborhood of $0 \in \R$. We describe another proof of his
characterization and some new development of the subject.Comment: 8 page

Strelcyn, Jean-Marie

English

arXiv

Recently A.R.Chouikha gave a new characterization of isochronicity of center at the origin for the equation x'' + g(x) = 0, where g is a real
smooth function de_ned in some neighborhood of 0 ϵ R. We present some
new development of the subject. The present text is a short account of my
paper "On Chouikha's isochronicity criterion", arXiv:1201.6503, where the
proofs can be found. We correct the formulation of some results from the
above paper

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

Analytic and Algebraic Geometry Lo´dz´ University Press 2013, 135 – 139ABOUT CHOUIKHA’S ISOCHRONICITY CRITERIONJEAN-MARIE STRELCYNAbstract. Recently A.R.Chouikha gave a new characterization of isochronic-ity of center at the origin for the equation x′′ + g(x) = 0, where g is a realsmooth function defined in some neighborhood of 0 ∈ R. We present somenew development of the subject. The present text is a short account of mypaper ”On Chouikha’s isochronicity criterion”, arXiv:1201.6503, where theproofs can be found. We correct the formulation of some results from theabove paper.Let us consider the second order differential equation(1) x′′ + g(x) = 0where g is a real function defined in some neighborhood of 0 ∈ R such that g(0) = 0,or equivalently the planar system(2)x˙ = yy˙ = −g(x)}.In what follows we shall exclusively concentrate on the system (2) with function gat least of class C1.As g(0) = 0, 0 ∈ R2 is a singular point of the system (2). If in some neighbor-hood of a singular point all orbits of the system are closed and surround it, thenthe singular point is called a center.A center is called isochronous if the periods of all orbits in some neighborhoodof it are constant.In future when speaking about isochronicity we always understand it with re-spect to 0 ∈ R2 and the system (2).The problem of characterization of isochronicity of the system (2) at 0 ∈ R2 interm of function g is an old one.2010 Mathematics Subject Classification. Primary 34C15, 34C25, 34C37.Key words and phrases. Center, isochronicity, Urabe function.135http://dx.doi.org/10.18778/7969-017-6.10136 JEAN-MARIE STRELCYNTo the best of our knowledge the first such characterization was done in 1937by I.Kukles and N.Piskunov in [3], where even the case of continuous functions gis considered. The second one was described in 1962 by M.Urabe in [5] (see also[4]). Unfortunately these characterizations are not easy to handle and they are notreally explicit.We shall denote(3) G(x) =∫ x0g(u) du.Let us denote by X the continuous function defined in some neighborhood of0 ∈ R by(4) (X(x))2 = 2G(x) and xX(x) > 0 for x 6= 0.Let us formulate now Urabe Isochronicity Criterion.Theorem 1 ([5]). Let g be a C1 function defined in some neighborhood of 0 ∈ R.Let g(0) = 0 and g′(0) = λ2, λ > 0. Then 0 ∈ R2 is an isochronous center for thesystem (2) if and only if(5) g(x) = λX(x)1 + h(X(x))where the function X is defined by (4) and where h is a continuous odd functiondefined in some neighborhood of 0 ∈ R.The function h is called Urabe function of the system (2).Let us note that ω =2piλis the period of orbits of the above isochronous center.Let us stress that from (3) and from assumptions on g in Urabe theorem itfollows that G(0) = 0 and that in some punctured neighborhood of 0, G(x) > 0, Gis of class C2. Under our assumptions one proves that X is of class C1. In fact, ifg ∈ Ck, k ≥ 1 (resp. g is real-analytic), then X is of class Ck, X ′(0) = λ > 0 andh is of class Ck−1 (resp. X and h are real-analytic).From now on we shall always assume that g ∈ C1(]− , [)) for some  > 0 andthatg′(0) = λ2, λ > 0.In September 2011 in a highly important paper [1], A.R.Chouikha publisheda completely new criterion of isochronicity ([1], Theorem B) which is much moredirect and explicit that all previously known.Theorem 2 ([1]). Let g ∈ C1(]− , [) for some  > 0. Let g(0) = 0 and g′(0) > 0.If there exists δ, 0 < δ ≤ , such that for |x| ≤ δ one has(6)ddx[G(x)g2(x)]= f(G(x))ABOUT CHOUIKHA’S ISOCHRONICITY CRITERION 137where f is a continuous functions defined on some interval [ 0, η ], where η > 0,then 0 ∈ R2 is an isochronous center for the system (2).If g ∈ C2(]− , [) and 0 ∈ R2 is an isochronous center for the system (2), thenthe condition (6) is satisfied.Consequently, if g ∈ C2(] − , [), then 0 ∈ R2 is an isochronous center for thesystem (2) if and only if the condition (6) is satisfied.We shall call the equation (6) the Chouikha equation and the function f is calledChouikha function of the system (2).Let us pause now in the history of this theorem. In early February 2010, A.R.Chouikha communicated to me his first proof of his theorem valid only in real-analytic setting. Some time after he presented to me the second proof also validonly in real-analytic setting. The first proof was based on Urabe theorem, thesecond one on S.N. Chow and D. Wang [2] formula for the derivative of the firstreturn map for the system (2). These proofs were not published at the time.At the beginning of July 2011, A.R. Chouikha and myself, simultaneously andindependently obtained two different proofs of Chouikha theorem in smooth setting.Both proofs are the adaptation of the previous Chouikha’s proofs in real-analyticsetting. The Chouikha’s proof published in [1] is the adaptation of his second proof.My proof is the adaptation of his first proof.As a consequence of this last proof we obtain an unexpected closed relationbetween Urabe function h and Chouikha function f .Theorem 3.(7) h(s) = λ∫ s0f(q22)dq,where g′(0) = λ2, λ > 0. Thusf is real-analytic (resp. of class C∞) if and only ifh is real-analytic (resp. of class C∞).From now on we shall suppose that f ∈ C1([0, ]),  > 0, where in 0 and in  oneconsiders the one-sided first derivatives. As before g ∈ C1(]− δ, δ[), δ > 0.Theorem 4. Let  > 0 and λ > 0. Let f ∈ C1([0, ]). There exists δ, 0 < δ ≤ and a unique function g ∈ C1(] − δ, δ[), g′(0) = λ2 such that for every |x| < δ theChouikha equation (6)ddx[G(x)g2(x)]= f(G(x))is satisfied.Let us stress that if f1, f2 ∈ C1([0, ]),  > 0, and f1 6= f2 on every interval[0, η], 0 < η ≤ , then in any neighborhood of 0 ∈ R, g1 6= g2, where g1 and g2 arethe solutions of Chouikha equation that correspond to f1 and to f2 respectively.Let us also note that if one supposes that f ∈ Ck([0, ]), 1 ≤ k ≤ ∞, or f isreal-analytic, then the unique solution g of Chouikha equation is also of the same138 JEAN-MARIE STRELCYNclass. This gives a new light on the matter of Sec.4 of [1], proving the convergenceof power series which appear there.From now on we shall only consider the case of real-analytic or C∞ functions g.Let us suppose that for function g, 0 ∈ R2 is an isochronous center for the system(2).In the real-analytic case there exists a natural bijective correspondence betweenthe set of the couples of real-analytic functions f defined in some neighborhood of0 ∈ R and of real numbers λ > 0 with the set of the real-analytic functions g suchthat 0 ∈ R2 is an isochronous center for the system (2). Indeed, to real-analyticfunction f defined in some neighborhood of 0 ∈ R and to real number λ > 0 weassociate the unique real-analytic function g such that g(0) = 0, g′(0) = λ2 which isa solution of Chouikha equation, the existence of which is given by Theorem 4. Letus stress that the completely analogous statement is valid also in C∞ framework.As a consequence of Theorem 4 and of Theorem 3 we obtain a fact that seemsto have been completely overlooked until now.Theorem 5. To every odd real-analytic (resp. of class C∞) function h defined insome neighborhood of 0 ∈ R and to every real number λ > 0 there corresponds aunique real-analytic (resp. of class C∞) function g defined in some neighborhoodof 0 ∈ R, g(0) = 0, g′(0) = λ2 such that 0 ∈ R2 is an isochronous center for thesystem (2) and that h is its Urabe function.Let us denote by Isochr(0, ω) the germs of isochronous centers of the equationx′′ + g(x) = 0 where g is a real-analytic function defined in some neighborhoodof 0 ∈ R, g(0) = 0, g′(0) > 0. Let us denote by Cω0 the germs of real-analyticfunctions defined in some neighborhood of 0 ∈ R. We can then state:Theorem 6. The Cartesian product Cω0 ×{x ∈ R;x > 0} and the set Isochr(0, ω)are in natural bijective correspondence. In other words the germs of real-analyticfunctions defined in some neighborhood of 0 ∈ R and the strictly positive real num-bers parametrize the germs of isochronous centers at 0 of equation x′′ + g(x) = 0,with g a real-analytic function defined in some neighborhood of 0 ∈ R, g(0) =0, g′(0) > 0.Let us stress that the completely analogous statement to Theorem 6 is valid alsoin C∞ framework.AcknowledgmentI thank A.Raouf Chouikha (University Paris 13) for many years of helpful dis-cussions on isochronous centers. I thank also Andrzej Maciejewski and MariaPrzybylska (both from University of Zielona Gora, Poland) and Alain Albouy (Ob-servatoire de Paris) for interesting discussions and important advices. Last but notleast I thank Marie-Claude Werquin (University Paris 13) for linguistic corrections.ABOUT CHOUIKHA’S ISOCHRONICITY CRITERION 139References[1] A.R. Chouikha. Period function and characterizations of isochronous potentials. arXiv:1109.4611, 2011.[2] S.N. Chow and D. Wang. On the monotonicity of the period function of some second orderequations. Casopis Pest. Mat.,111, 14-25, 1986.[3] I. Koukles and N. Piskounov. Sur les vibrations tautochrones dans les systmes conservatifs etnon conservatifs. C. R. Acad. Sci., URSS, XVII(9), 417-475, 1937.[4] M. Robnik and V.G. Romanovskii. On Urabe’s criteria of isochronicity. J. Phys; A 32(7),1279-1283, 1999.[5] M. Urabe. The potential force yielding a periodic motion whose period is an arbitrary contin-uous function of the amplitude of the velocity. Arch. Ration. Mech. Anal., 11:27–33, 1962.Laboratoire de Mathmatiques Raphae¨l Salem,, CNRS, Universite´ de Rouen, Avenuede l’Universit BP 12, 76801 Saint-Etienne-du-Rouvray, FranceLaboratoire Analyse Ge´ometrie et Applications,, UMR CNRS 7539, Institut Gallile,Universit Paris 13,, 99 Avenue J.-B. Clment, 93430 Villetaneuse, FranceE-mail address: strelcyn@math.univ-paris13.fr

On Chouikha’s isochronicity criterion

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On Chouikha's isochronicity criterion

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