# An inverse approach to the center problem

Article

## Abstract

We consider analytic or polynomial vector fields of the form $${\mathcal {X}}=\left( -y+X\right) \dfrac{\partial }{\partial x}+\left( x+Y\right) \dfrac{\partial }{\partial y},$$ where $$X=X(x,y))$$ and $$Y=Y(x,y))$$ start at least with terms of second order. It is well-known that $${\mathcal {X}}$$ has a center at the origin if and only if $${\mathcal {X}}$$ has a Liapunov–Poincaré local analytic first integral of the form $$H=\dfrac{1}{2}(x^2+y^2)+\sum _{j=3}^ {\infty } H_j$$, where $$H_j=H_j(x,y)$$ is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of $${\mathcal {X}}$$ is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which H is a first integral. Moreover, given an analytic function $$V=1+\sum _{j=1}^ {\infty } V_j$$ in a neighborhood of the origin, where $$V_j$$ is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form $$H=\dfrac{1}{2}(x^2+y^2)\,\left( 1+ \sum _{j=1}^{\infty } \Upsilon _j\right) ,$$ in a neighborhood of the origin, where $$\Upsilon _j$$ is a homogenous polynomial of degree j for $$j\ge 1.$$ These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.

## Keywords

Center-focus problem Analytic planar differential system Liapunov’s constants Isochronous center Darboux’s first integral Weak condition for a center Weak center

34C05 34C07

## References

1. 1.
Poincaré, H.: Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II. Rendiconti del circolo matematico di Palermo 5, 161–191 (1891)
2. 2.
Poincaré, H.: Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II. Rendiconti del circolo matematico di Palermo 11, 193–239 (1897)
3. 3.
Dulac, H.: Détermination et integration d’ une classe d’ equations differentielle ayant pour point singulier un centre. Bull. Sci. Math Sér. 32, 230–252 (1908)
4. 4.
Bendixson, I.: Sur les courbes définies par des equations differentielles. Acta Math. 24, 1–88 (1901)
5. 5.
Frommer, M.: Die Integralkurven einer gewohnlichen Di?erential-gleichung erster Ordnung in der Umgebung rationaler Unbestimmtheitsstellen. Math. Ann. 99, 222–272 (1928)
6. 6.
Liapounoff, M.A.: Problème général de la stabilité du mouvement, Annals of Mathematics Studies 17. Princeton University Press, Princeton (1947)
7. 7.
Llibre, J., Ramírez, R., Ramírez, V.: An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities. J. Differ. Equ. 263, 3327–3369 (2016)
8. 8.
Malkin, I.G.: Criteria for center of a differential equation. Volg. Matem. Sb. 2, 87–91 (1964). (in Russian)
9. 9.
Ilyashenko, Y., Yakovenko, S.: Lectures on Analytic Differential Equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI (2008)
10. 10.
Malkin, I.G.: Stability Theory of Movements. Nauka, Moscow (1966). (in Russian)Google Scholar
11. 11.
Bautin, N.N.: On the number of limit cycles appearing with variation of the coefficients from an equilibrium state of the type of a focus or a center, Mat. Sb. (N.S.) 30(72), 181–196 (1952)
12. 12.
Saharnikov, N.A.: Solution of the center focus problem in one case. Prikl. Mat. Meh. 14, 651–658 (1950)
13. 13.
Sibirskii, K.S.: Method of Invariants in the Qualitative Theory of Differential Equations. Acad. Sci. Moldavian SSR, Kishinev (1968). (in Russian)Google Scholar
14. 14.
Reeb, G.: Sur certaines propiétés topologiques des variétés feuilletées , Wu, W.T., Reeb, G. (eds.), Sur les espaces fibrés et les variétés feuilletées, Tome XI, In: Actualités Sci. Indust. vol. 1183, Hermann et Cie, Paris (1952)Google Scholar
15. 15.
Darboux, G.: Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges). Bull. Sci. math. 2ème série 2, 60–96; 123–144; 151–200 (1878)Google Scholar
16. 16.
Christopher, C., Li, C.: Limit Cycles of Differential Equations, Advances Courses in Mathematics, CRM Barcelona. Birkhäuser Verlag, Basel (2007)Google Scholar
17. 17.
Llibre, J.: Integrability of polynomial differential systems. In: Cañada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, Ordinary Differential Equations, pp. 437–533. Elsevier, Amsterdam (2004)
18. 18.
Zoladek, H.: The solution of the center-focus problem. Institute of Mathematics, University of Warsaw, Poland (1992)Google Scholar
19. 19.
Mikonenko, V.: A reflecting function of a family of functions. Differentsial’nye uravneniya 36, 1636–1641 (1989)
20. 20.
Llibre, J., Ramírez, R.: Inverse Problems in Ordinary Differential Equations and Applications, Progress in Mathematics, vol. 313. Birkhäuser, Basel (2016)
21. 21.
Sadovskaia, N.: Inverse problem in theory of ordinary differential equations. Thesis Ph.D., Univ. Politécnica de Cataluña (2002) (in Spanish)Google Scholar
22. 22.
Belitskii, G.: Smooth equivalence of germs of $$C^\infty$$ of vector fields with one zero or a pair of pure imaginary eigenvalues. Funct. Anal. Appl. 20(4), 253–259 (1986)
23. 23.
Alwash, M.A., Lloyd, N.G.: Non-autonomous equations related to polynomial two dimensional systems. Proc. R. Soc. Edinb. 105 A, 129–152 (1987)
24. 24.
Conti, R.: Centers of planar polynomial systems. a review. Le Matematiche Vol. LIII, Fasc. II, 207–240 (1998)Google Scholar
25. 25.
Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960)
26. 26.
Devlin, J.: Coexisting isochronous centers and non-isochronous centers. Bull. Lond. Math. 98, 495–500 (1996)
27. 27.
Lukashevich, N.: Isochronism of the ecenter of certain systems of differential equations. Diff. Uravn. 1, 295–302 (1965). (in Russian)Google Scholar
28. 28.
Pleshkan, I.I.: A new method of investigating the isochronism of a system of two differential equations. Diff. Uravn. 5, 1083–1090 (1969). (in Russian)
29. 29.
Sternberg, S.: Lectures on Differential Geometry. Prentice Hall, Englewood Cliffs (1964)
30. 30.
Llibre, J., Ramírez, R., Ramírez,V., Sadovskaia, N.: Centers and uniform isochronous centers of planar polynomial differential systems. Preprint (2017)Google Scholar
31. 31.
Mardeâić, P., Rousseau, C., Toni, B.: Linearization of isochronous center. J. Differ. Equ. 121, 67–108 (1995)
32. 32.
Chavarriga, J., Sabatini, M.: A survey on isochronous center. Qualit. Theory Dyn. Syst. 1, 1–70 (1999)
33. 33.
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems, Universitext. Springer, Berlin (2006)
34. 34.
Giné, J., Grau, M., Llibre, J.: On the extensions of the Darboux theory of integrability. Nonlinearity 26, 2221–2229 (2013)
35. 35.
Zhou, Z.: A new method for research on the center-focus problem of differential systems. Abstr. Appl. Anal. 2014, 1–5 (2014)

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## Authors and Affiliations

• Jaume Llibre
• 1
• Rafael Ramírez
• 2
• Valentín Ramírez
• 3
1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
3. 3.Universitat Central de BarcelonaBarcelonaSpain