Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T09:29:14.175Z Has data issue: false hasContentIssue false

Universal curves in the center problem for Abel differential equations

Published online by Cambridge University Press:  15 December 2014

ALEXANDER BRUDNYI*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada email [email protected]

Abstract

We study the center problem for the class ${\mathcal{E}}_{{\rm\Gamma}}$ of Abel differential equations $dv/dt=a_{1}v^{2}+a_{2}v^{3}$, $a_{1},a_{2}\in L^{\infty }([0,T])$, such that images of Lipschitz paths $\tilde{A}:=(\int _{0}^{\cdot }a_{1}(s)\,ds,\int _{0}^{\cdot }a_{2}(s)\,ds):[0,T]\rightarrow \mathbb{R}^{2}$ belong to a fixed compact rectifiable curve ${\rm\Gamma}$. Such a curve is said to be universal if whenever an equation in ${\mathcal{E}}_{{\rm\Gamma}}$ has center on $[0,T]$, this center must be universal, i.e. all iterated integrals in coefficients $a_{1},a_{2}$ of this equation must vanish. We investigate some basic properties of universal curves. Our main results include an algebraic description of a universal curve in terms of a certain homomorphism of its fundamental group into the group of locally convergent invertible power series with the product being the composition of series, explicit examples of universal curves, and approximation of Lipschitz triangulable curves by universal ones.

Type
Research Article
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alwash, M. A. M.. On the composition conjectures. Electron. J. Differential Equations 2003(69) (2003), 14.Google Scholar
Alwash, M. A. M.. The composition conjecture for Abel equation. Expo. Math. 27 (2009), 241250.Google Scholar
Alwash, M. A. M.. Polynomial differential equations with piecewise linear coefficients. Differ. Equ. Dyn. Syst. 19(3) (2011), 267281.CrossRefGoogle Scholar
Alwash, M. A. M.. Composition conditions for two-dimensional polynomial systems. Differ. Equ. Appl. 5(1) (2013), 112.Google Scholar
Alwash, M. A. M. and Lloyd, N. G.. Non-autonomous equations related to polynomial two-dimensional systems. Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 129152.Google Scholar
Brudnyi, A.. An explicit expression for the first return map in the center problem. J. Differential Equations 206 (2004), 306314.Google Scholar
Brudnyi, A.. On the center problem for ordinary differential equations. Amer. J. Math. 128(2) (2006), 419451.Google Scholar
Brudnyi, A.. Formal paths, iterated integrals and the center problem for ordinary differential equations. Bull. Sci. Math. 132 (2008), 455485.Google Scholar
Brudnyi, A.. Center problem for ODEs with coefficients generating the group of rectangular paths. C. R. Math. Acad. Sci. Soc. R. Can. 31(2) (2009), 3344.Google Scholar
Brudnyi, A.. Free subgroups of the group of formal power series and the center problem for ODEs. C. R. Math. Acad. Sci. Soc. R. Can. 31(4) (2009), 97106.Google Scholar
Brudnyi, A.. Composition conditions for classes of analytic functions. Nonlinearity 25 (2012), 31973209.Google Scholar
Brudnyi, A.. Moments finiteness problem and characterization of universal centers of ODEs with analytic coefficients. Nonlinearity 27 (2014), 16111631.Google Scholar
Briskin, M., Francoise, J.-P. and Yomdin, Y.. The Bautin ideal of the Abel equation. Nonlinearity 11 (1998), 4153.Google Scholar
Briskin, M., Francoise, J.-P. and Yomdin, Y.. Center conditions, compositions of polynomials and moments on algebraic curves. Ergod. Th. & Dynam. Sys. 19 (1999), 12011220.CrossRefGoogle Scholar
Blinov, M.. Center and composition conditions for Abel equation. PhD Thesis, Weizmann Institute of Science, 2002.Google Scholar
Blinov, M., Roytvarf, N. and Yomdin, Y.. Center and moment conditions for Abel equation with rational coefficients. Funct. Differ. Equ. 10(1–2) (2003), 95106.Google Scholar
Briskin, M., Roytvarf, N. and Yomdin, Y.. Center conditions at infinity for Abel differential equations. Ann. of Math. (2) 172(1) (2010), 437483.Google Scholar
Brudnyi, A. and Yomdin, Y.. Tree composition condition and moments vanishing. Nonlinearity 23 (2010), 16511673.Google Scholar
Christopher, C.. Abel equations: composition conjectures and the model problem. Bull. Lond. Math. Soc. 32 (2000), 332338.Google Scholar
Cima, A., Gasull, A. and Mãnosas, F.. Centers for trigonometric Abel equations. Qual. Theory Dyn. Syst. 11 (2012), 1937.Google Scholar
Cima, A., Gasull, A. and Mãnosas, F.. A simple solution of some composition conjectures for Abel equations. J. Math. Anal. Appl. 398 (2013), 477486.Google Scholar
Cima, A., Gasull, A. and Mãnosas, F.. An explicit bound of the number of vanishing double moments forcing composition. J. Differential Equations 255(3) (2013), 339350.Google Scholar
Cohen, S. D.. The group of translations and positive rational powers is free. Q. J. Math. 46(1) (1995), 2193.CrossRefGoogle Scholar
Giné, J., Grau, M. and Llibre, J.. Universal centers and composition conditions. Proc. Lond. Math. Soc. 106(3) (2013), 481507.Google Scholar
Giné, J., Grau, M. and Santallusia, X.. Universal centers in the cubic trigonometric Abel equation. Electron. J. Qual. Theory Differ. Equ.(1) (2014), 17.Google Scholar
Hu, S.-T.. Homotopy Theory. Academic Press, New York, 1959.Google Scholar
Il’yashenko, Yu.. Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. (N.S.) 39(3) (2002), 301354.Google Scholar
Muzychuk, M. and Pakovich, F.. Solution of the polynomial moment problem. Proc. Lond. Math. Soc. 99(3) (2009), 633657.Google Scholar
Pakovich, F.. On the polynomial moment problem. Math. Res. Lett. 10(2–3) (2003), 401410.Google Scholar
Pakovich, F.. On rational functions orthogonal to all powers of a given rational function on a curve. Moscow Math. J. 13(4) (2013), 693731.CrossRefGoogle Scholar
Pakovich, F.. Solution of the parametric center problem for the Abel differential equation. Preprint, 2014, arXiv:1407.0150.Google Scholar
Pakovich, F., Roytvarf, N. and Yomdin, Y.. Cauchy type integrals of algebraic functions. Israel J. Math. 144 (2004), 221291.Google Scholar