# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022095
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## A proof of a Dumortier-Roussarie's conjecture

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 School of Mathematics (Zhuhai), Sun Yat-sen University, 519086 Zhuhai, China

* Corresponding author: Changjian Liu

Received  December 2021 Revised  March 2022 Early access April 2022

Fund Project: The authors are partially supported by the National Natural Science Foundation of China (Grant No. 11771282 and No. 12171491)

Dumortier and Roussarie proposed a conjecture in their paper (2009, Discrete Con. Dyn. Sys., 2,723-781): For any $q\in {\mathbb{N}}$, the Abelian integrals $J_{2j+1}(h) = \int_{\gamma_h}x^{2j-1}\,\mathrm dy$, $j = 0, 1, 2, \cdots, q$, form a strict Chebyshev system on intervals $h\in (0, \frac{1}{2}]$, where $\gamma_h = \{(x, y)| \mathrm e^{-2y}(y+\frac{1}{2}-x^2) = h\}$. If this conjecture holds, then they obtain the precise upper bound of the number of limit cycles that appear near a slow-fast Hopf point of any codimension. In the present paper we develop a method to estimate the number of zeros of Abelian integrals and prove this conjecture.

Citation: Chengzhi Li, Changjian Liu. A proof of a Dumortier-Roussarie's conjecture. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022095
##### References:
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show all references

##### References:
 [1] V. I. Arnol'd, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funkcional. Anal. i Priložen, 11 (1977), 85-92. [2] V. I. Arnol'd, Ten problems, Adv. Soviet Math., 1 (1990), 1-8. [3] C. Christopher and C. Li, Limit Cycles in Differential Equations, Birkhäuser Verlag, Basel, 2007. [4] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Spring-Verlag, 2006. [5] F. Dumortier and R. Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 723-781.  doi: 10.3934/dcdss.2009.2.723. [6] J.-L. Figuerasa, W. Tucker and J. Villadelprat, Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals, J. Differential Equations, 254 (2013), 3647-3663.  doi: 10.1016/j.jde.2013.01.036. [7] J.-P. Françcoise and D. Xiao, Perturbation theory of a symmetric center within Liénard equations, J. Differential Equations, 259 (2015), 2408-2429.  doi: 10.1016/j.jde.2015.03.039. [8] M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X. [9] I. D. Iliev, Perturbation of quadratic centers, Bull. Sci. math., 122 (1998), 107-161.  doi: 10.1016/S0007-4497(98)80080-8. [10] J. Karlin and W. J. Studden, T-Systems: With Applications in Analysis and Statistics, Pure Appl. Math., Interscience Publishers, New York, London, Sidney, 1966. [11] C. Liu, G. Chen and Z. Sun, New criteria for the monotonicity of the ratio of two Abelian integrals, J. Math. Anal. Appl., 465 (2018), 220-234.  doi: 10.1016/j.jmaa.2018.04.074. [12] C. Liu, C. Li and J. Llibre, The cyclicity of the period annulus of a reversible quadratic system, Proceedings of the Royal Society of Edinburgh, published on-line. [13] C. Liu and D. Xiao, The lowest upper bound on the number of zeros of Abelian Integrals, J. Differential Equations, 269 (2020), 3816-3852.  doi: 10.1016/j.jde.2020.03.016. [14] D. Marin and J. Villadelprat, On the Chebyshev property of certain Abelian integrals near a polycycle, Qual. Theory Dyn. Syst., 17 (2018), 261-270.  doi: 10.1007/s12346-017-0226-3.
The phase portrait of system (3) in the Poincaré disc
The orbit $\gamma_h$ when $h_n<h<h_{n+1}$
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