Center conditions at infinity for Abel differential equations. (English) Zbl 1216.34025

Briskin, Miriam; Roytvarf, Nina; Yomdin, Yosef

Ann. Math. (2) 172, No. 1, 437-483 (2010).

It is said that the Abel differential equation
\[ y'=p(x) y^2+q(x) y^3,\tag{1} \]
where \(p\) and \(q\) are polynomials, has a center at a set \(A=\{a_1,\dots, a_r\}\) of complex numbers if \(y(a_1)=\dots=y(a_r)\) for any solution \(y(x)\) with the initial value \(y(a_1)\) sufficiently small. The center problem for (1) is to give necessary and sufficient conditions for the Abel equation (1) to have a center in terms of \(p\) and \(q\). Although the original interest in the center problem for the Abel equation comes from the study of the center problem for polynomial vector fields (since some of these fields can be transformed to the Abel equation above), there is no doubt that the problem is important and interesting also in its own right.
The polynomials \(p\) and \(q\) are said to satisfy the Polynomial Composition Conjecture (PCC) on \(A\) if there exist polynomials \(\widetilde P\), \(\widetilde Q\) and \(W\) such that \(P=\int p\) and \(Q=\int q\) are representable as \( P(x)=\widetilde P(W(x))\), \(Q(x)=\widetilde Q(W(x)), \) and \( W(a_1)=\dots=W(a_r)\).
The PCC is a sufficient condition for the center problem and it is known also to be necessary for the center problem for small degrees of \(p\) and \(q\) and in some other very special situations.
The authors show that, for wide ranges of degrees of \(P\) and \(Q\) (restricted only by certain assumptions on the common divisors of their degrees), the composition condition provides a very accurate approximation of the center one – up to a finite number of configurations not accounted for. It appears to be the first general (that is, not restricted to small degrees of \(p\) and \(q\) or to a very special form of these polynomials) result regarding the center problem for Abel equations.
As an important intermediate result, it is shown that “at infinity” (according to an appropriate projectivization of the parameter space) the center conditions are given by a system of the moment equations of the form \(\int^{a_s}_{a_1} P^k q = 0\), \(s=2,\dots,r\), \(k=0,1,\dots\)

Reviewer: Valery Romanovski (Maribor)

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07	Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08	Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C25	Periodic solutions to ordinary differential equations
30E05	Moment problems and interpolation problems in the complex plane
34M35	Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms

Keywords:

center-focus problem; Abel differential equation; polynomial composition conjecture; generalized moment problem

References:

[1]	M. A. M. Alwash and N. G. Lloyd, ”Nonautonomous equations related to polynomial two-dimensional systems,” Proc. Roy. Soc. Edinburgh, vol. 105A, pp. 129-152, 1987. · Zbl 0618.34026
[2]	M. A. M. Alwash, ”On a condition for a centre of cubic nonautonomous equations,” Proc. Roy. Soc. Edinburgh, vol. 113A, iss. 3-4, pp. 289-291, 1989. · Zbl 0685.34025
[3]	M. A. M. Alwash, ”Word problems and the centers of Abel differential equations,” Ann. Differential Equations, vol. 11, iss. 4, pp. 392-396, 1995. · Zbl 0684.34025
[4]	M. A. M. Alwash, ”Periodic solutions of a quartic differential equation and Groebner bases,” J. Comput. Appl. Math., vol. 75, iss. 1, pp. 67-76, 1996. · Zbl 0864.34035
[5]	M. A. M. Alwash, ”On the center conditions of certain cubic systems,” Proc. Amer. Math. Soc., vol. 126, iss. 11, pp. 3335-3336, 1998. · Zbl 0903.34026
[6]	M. A. M. Alwash, ”On the composition conjectures,” Electron. J. Differential Equations, p. 69, 2003. · Zbl 1054.34047
[7]	V. Arnol’d, ”Problems on singularities and dynamical systems,” in Developments in Mathematics: The Moscow School, London: Chapman & Hall, 1993, pp. 251-274. · Zbl 0883.58016
[8]	V. Arnol’d and Y. Il’yashenko, ”Ordinary differential equations,” in Dynamical Systems. I, New York: Springer-Verlag, 1988, vol. 1, p. x. · Zbl 0471.70025
[9]	M. Blinov, ”Some computations around the center problem, related to the composition algebra of univariate polynomials,” PhD Thesis , Weizmann Institute of Science, 1997.
[10]	M. Blinov, ”Center and Composition conditions for Abel Equation,” PhD Thesis , Weizmann Institute of Science, 2002.
[11]	M. Blinov, M. Briskin, and Y. Yomdin, ”Local center conditions for the Abel equation and cyclicity of its zero solution,” in Complex Analysis and Dynamical Systems II, Providence, RI: Amer. Math. Soc., 2005, vol. 382, pp. 65-82. · Zbl 1095.34018
[12]	M. Briskin, ”Infinitesimal center-focus problem, generalized moments and composition of polynomials,” Funct. Differ. Equ., vol. 6, iss. 1-2, pp. 47-53, 1999. · Zbl 1043.34025
[13]	M. Briskin, Algebra of generalized moments and composition of polynomials, 2000. · Zbl 1043.34025
[14]	M. Briskin, Recursive moments representation and quantitative moment problem. · Zbl 1043.34025
[15]	M. Briskin, J. -P. Fran\ccoise, and Y. Yomdin, ”Une approche au problème du centre-foyer de Poincaré,” C. R. Acad. Sci. Paris Sér. I Math., vol. 326, iss. 11, pp. 1295-1298, 1998. · Zbl 0914.34025
[16]	M. Briskin, J. -P. Fran\ccoise, and Y. Yomdin, ”Center conditions, compositions of polynomials and moments on algebraic curves,” Ergodic Theory Dynam. Systems, vol. 19, iss. 5, pp. 1201-1220, 1999. · Zbl 0990.34017
[17]	M. Briskin, J. -P. Fran\ccoise, and Y. Yomdin, ”Center conditions. II. Parametric and model center problems,” Israel J. Math., vol. 118, pp. 61-82, 2000. · Zbl 0989.34021
[18]	M. Briskin, J. -P. Fran\ccoise, and Y. Yomdin, ”Center conditions. III. Parametric and model center problems,” Israel J. Math., vol. 118, pp. 83-108, 2000. · Zbl 0989.34022
[19]	M. Briskin, J. -P. Fran\ccoise, and Y. Yomdin, ”Generalized moments, center-focus conditions, and compositions of polynomials,” in Operator Theory, System Theory and Related Topics, Basel: Birkhäuser, 2001, pp. 161-185. · Zbl 1075.34509
[20]	M. Briskin and Y. Yomdin, ”Tangential version of Hilbert 16th problem for the Abel equation,” Mosc. Math. J., vol. 5, iss. 1, pp. 23-53, 2005. · Zbl 1097.34025
[21]	A. Brudnyi, ”An algebraic model for the center problem,” Bull. Sci. Math., vol. 128, iss. 10, pp. 839-857, 2004. · Zbl 1084.34035
[22]	A. Brudnyi, ”An explicit expression for the first return map in the center problem,” J. Differential Equations, vol. 206, iss. 2, pp. 306-314, 2004. · Zbl 1066.34026
[23]	A. Brudnyi, ”On the center problem for ordinary differential equations,” Amer. J. Math., vol. 128, iss. 2, pp. 419-451, 2006. · Zbl 1104.34022
[24]	A. Brudnyi, ”On center sets of ODEs determined by moments of their coefficients,” Bull. Sci. Math., vol. 130, iss. 1, pp. 33-48, 2006. · Zbl 1102.34019
[25]	A. Brudnyi, Private communication.
[26]	K. T. Chen, ”Iterated path integrals,” Bull. Amer. Math. Soc., vol. 83, iss. 5, pp. 831-879, 1977. · Zbl 0389.58001
[27]	L. A. vCerkas, ”The number of limit cycles of a certain second order autonumous system,” Differencial’ nye Uravnenija, vol. 12, iss. 5, pp. 944-946, 960, 1976.
[28]	C. Christopher, ”An algebraic approach to the classification of centers in polynomial Liénard systems,” J. Math. Anal. Appl., vol. 229, iss. 1, pp. 319-329, 1999. · Zbl 0921.34033
[29]	C. Christopher, ”Abel equations: composition conjectures and the model problem,” Bull. London Math. Soc., vol. 32, iss. 3, pp. 332-338, 2000. · Zbl 1047.34019
[30]	C. Christopher and C. Li, Limit Cycles of Differential Equations, Basel: Birkhäuser, 2007. · Zbl 1359.34001
[31]	C. B. Collins, ”Conditions for a centre in a simple class of cubic systems,” Differential Integral Equations, vol. 10, iss. 2, pp. 333-356, 1997. · Zbl 0894.34022
[32]	J. Devlin, ”Word problems related to periodic solutions of a nonautonomous system,” Math. Proc. Cambridge Philos. Soc., vol. 108, iss. 1, pp. 127-151, 1990. · Zbl 0726.34026
[33]	J. Devlin, ”Word problems related to derivatives of the displacement map,” Math. Proc. Cambridge Philos. Soc., vol. 110, iss. 3, pp. 569-579, 1991. · Zbl 0744.34032
[34]	J. Devlin, N. G. Lloyd, and J. M. Pearson, ”Cubic systems and Abel equations,” J. Differential Equations, vol. 147, iss. 2, pp. 435-454, 1998. · Zbl 0911.34020
[35]	J. P. Francoise, ”Successive derivatives of a first return map, application to the study of quadratic vector fields,” Ergodic Th. Dynam. Syst., vol. 16, iss. 1, pp. 87-96, 1996. · Zbl 0852.34008
[36]	J. -P. Fran\ccoise, ”Local bifurcations of limit cycles, Abel equations and Liénard systems,” in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Dordrecht: Kluwer Acad. Publ., 2004, pp. 187-209.
[37]	J. -P. Francoise and Y. Yomdin, ”Bernstein inequalities and applications to analytic geometry and differential equations,” J. Funct. Anal., vol. 146, iss. 1, pp. 185-205, 1997. · Zbl 0869.34008
[38]	A. Gasull and J. Llibre, ”Limit cycles for a class of Abel equations,” SIAM J. Math. Anal., vol. 21, iss. 5, pp. 1235-1244, 1990. · Zbl 0732.34025
[39]	L. Gavrilov, ”Higher order Poincaré-Pontryagin functions and iterated path integrals,” Ann. Fac. Sci. Toulouse Math., vol. 14, iss. 4, pp. 663-682, 2005. · Zbl 1104.34024
[40]	L. Gavrilov and I. D. Iliev, ”Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems,” Ergodic Theory Dynam. Systems, vol. 20, iss. 6, pp. 1671-1686, 2000. · Zbl 0992.37054
[41]	L. Gavrilov and I. D. Iliev, ”The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields,” Amer. J. Math., vol. 127, iss. 6, pp. 1153-1190, 2005. · Zbl 1093.34015
[42]	J. Giné, ”The nondegenerate center problem and the inverse integrating factor,” Bull. Sci. Math., vol. 130, iss. 2, pp. 152-161, 2006. · Zbl 1102.34001
[43]	V. Golubitski and V. Guillemin, ”Stable mappings and their singularities,” Grad. Texts in Math., vol. 14, 1973. · Zbl 0294.58004
[44]	Y. Ilyashenko, ”Centennial history of Hilbert’s 16th problem,” Bull. Amer. Math. Soc., vol. 39, iss. 3, pp. 301-354, 2002. · Zbl 1004.34017
[45]	V. V. Ivanov and E. P. Volokitin, ”Uniformly isochronous polynomial centers,” Electron. J. Differential Equations, p. 133, 2005. · Zbl 1094.34019
[46]	A. Lins Neto, ”On the number of solutions of the equation \(dx/dt=\sum ^n_{j=0}\,a_j(t)x^j\), \(0\leq t\leq 1\), for which \(x(0)=x(1)\),” Invent. Math., vol. 59, iss. 1, pp. 67-76, 1980. · Zbl 0448.34012
[47]	N. G. Lloyd, ”The number of periodic solutions of the equation \(\dot z=z^N+p_1(t)z^{N-1}+\cdots +p_N(t)\),” Proc. London Math. Soc., vol. 27, pp. 667-700, 1973. · Zbl 0273.34042
[48]	H. Movasati, ”Center conditions: rigidity of logarithmic differential equations,” J. Differential Equations, vol. 197, iss. 1, pp. 197-217, 2004. · Zbl 1049.32033
[49]	F. Pakovich and M. Muzychuk, ”Solution of the polynomial moment problem,” Proc. Lond. Math. Soc., vol. 99, iss. 3, pp. 633-657, 2009. · Zbl 1177.30046
[50]	F. Pakovich, ”A counterexample to the “composition conjecture”,” Proc. Amer. Math. Soc., vol. 130, iss. 12, pp. 3747-3749, 2002. · Zbl 1008.34024
[51]	F. Pakovich, ”On the polynomial moment problem,” Math. Res. Lett., vol. 10, iss. 2-3, pp. 401-410, 2003. · Zbl 1043.30021
[52]	F. Pakovich, ”On polynomials orthogonal to all powers of a Chebyshev polynomial on a segment,” Israel J. Math., vol. 142, pp. 273-283, 2004. · Zbl 1058.30007
[53]	F. Pakovich, ”On polynomials orthogonal to all powers of a given polynomial on a segment,” Bull. Sci. Math., vol. 129, iss. 9, pp. 749-774, 2005. · Zbl 1089.30039
[54]	F. Pakovich, N. Roytvarf, and Y. Yomdin, ”Cauchy-type integrals of algebraic functions,” Israel J. Math., vol. 144, pp. 221-291, 2004. · Zbl 1078.30034
[55]	J. M. Pearson, N. G. Lloyd, and C. J. Christopher, ”Algorithmic derivation of centre conditions,” SIAM Rev., vol. 38, iss. 4, pp. 619-636, 1996. · Zbl 0876.34033
[56]	A. Poincaré, Oeuvres, Les Grands Classiques, Villars: Gauthier, 1995.
[57]	J. F. Ritt, ”Prime and composite polynomials,” Trans. Amer. Math. Soc., vol. 23, iss. 1, pp. 51-66, 1922. · JFM 48.0079.01
[58]	R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, Basel: Birkhäuser, 1998, vol. 164. · Zbl 0898.58039
[59]	N. Roytvarf, ”Generalized moments, composition of polynomials and Bernstein classes,” in Entire Functions in Modern Analysis, Ramat Gan: Bar-Ilan Univ., 2001, pp. 339-355. · Zbl 0996.30025
[60]	N. Roytwarf and Y. Yomdin, ”Bernstein classes,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 47, iss. 3, pp. 825-858, 1997. · Zbl 0974.30524
[61]	N. Roytvarf and Y. Yomdin, ”Analytic continuation of Cauchy-type integrals,” Funct. Differ. Equ., vol. 12, iss. 3-4, pp. 375-388, 2005. · Zbl 1072.30029
[62]	S. Shahshahani, ”Periodic solutions of polynomial first order differential equations,” Nonlinear Anal., vol. 5, iss. 2, pp. 157-165, 1981. · Zbl 0449.34027
[63]	A. Schinzel, Polynomials with Special Regard to Reducibility, Cambridge: Cambridge Univ. Press, 2000. · Zbl 0956.12001
[64]	D. Schlomiuk, ”Algebraic particular integrals, integrability and the problem of the center,” Trans. Amer. Math. Soc., vol. 338, iss. 2, pp. 799-841, 1993. · Zbl 0777.58028
[65]	K. S. Sibirsky, Introduction to the Algebraic Theory of Invariants of Differential Equations, Manchester: Manchester Univ. Press, 1988. · Zbl 0691.34032
[66]	S. Smale, ”Mathematical problems for the next century,” Math. Intelligencer, vol. 20, iss. 2, pp. 7-15, 1998. · Zbl 0947.01011
[67]	L. Yang and Y. Tang, ”Some new results on Abel equations,” J. Math. Anal. Appl., vol. 261, iss. 1, pp. 100-112, 2001. · Zbl 0995.34031
[68]	Y. Yomdin, ”Global finiteness properties of analytic families and algebra of their Taylor coefficients,” in The Arnoldfest, Providence, RI: Amer. Math. Soc., 1999, pp. 527-555. · Zbl 0944.34003
[69]	Y. Yomdin, ”The center problem for the Abel equation, compositions of functions, and moment conditions,” Moscow Math. J., vol. 3, iss. 3, pp. 1167-1195, 2003. · Zbl 1086.34031
[70]	Y. Yomdin and G. Comte, Tame Geometry with Application in Smooth Analysis, New York: Springer-Verlag, 2004. · Zbl 1076.14079
[71]	H. Zoladek, ”The problem of center for resonant singular points of polynomial vector fields,” J. Differential Equations, vol. 137, iss. 1, pp. 94-118, 1997. · Zbl 0885.34034
[72]	H. Zoladek, ”Asymptotic properties of abelian integrals arising in quadratic systems,” Bull. Belg. Math. Soc. Simon Stevin, vol. 7, iss. 2, pp. 265-276, 2000. · Zbl 1009.34030
[73]	H. Zoladek, Private communication.