doi: 10.3934/dcdss.2022095
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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:
[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. FiguerasaW. 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. GrauF. 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. LiuG. 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.

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. FiguerasaW. 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. GrauF. 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. LiuG. 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.

Figure 1.  The phase portrait of system (3) in the Poincaré disc
Figure 2.  The orbit $ \gamma_h $ when $ h_n<h<h_{n+1} $
[1]

Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175

[2]

Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control and Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022

[3]

Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259

[4]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[5]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447

[6]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279

[7]

Mario E. Chávez-Gordillo, Bernardo San Martín, Jaime Vera. Persistent singular attractors arising from singular cycle under symmetric conditions. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 671-685. doi: 10.3934/dcds.2011.30.671

[8]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[9]

Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818

[10]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

[11]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[12]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081

[13]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[14]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[15]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893

[16]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783

[17]

Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017

[18]

Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309

[19]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure and Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[20]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic and Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (132)
  • HTML views (50)
  • Cited by (0)

Other articles
by authors

[Back to Top]