American Institute of Mathematical Sciences

Advanced
December  2009, 2(4): 723-781. doi: 10.3934/dcdss.2009.2.723

Birth of canard cycles

Freddy Dumortier 1, and  Robert Roussarie 2,

1. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek
2. 

Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex

Received  September 2008 Revised  March 2009 Published  September 2009

In this paper we consider singular perturbation problems occuring in planar slow-fast systems $(\dot x=y-F(x,\lambda),\dot y=-\varepsilon G(x,\lambda))$ where $F$ and $G$ are smooth or even real analytic for some results, $\lambda$ is a multiparameter and $\varepsilon$ is a small parameter. We deal with turning points that are limiting situations of (generalized) Hopf bifurcations and that we call slow-fast Hopf points. We investigate the number of limit cycles that can appear near a slow-fast Hopf point and this under very general conditions. One of the results states that for any analytic family of planar systems, depending on a finite number of parameters, there is a finite upperbound for the number of limit cycles that can bifurcate from a slow-fast Hopf point.
   The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size. This explains the title of the paper. The treatment is based on blow-up, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slow-divergence integral as it is used in singular perturbation theory to Abelian integrals that have to be used in studying limit cycles close to the singular point.
Keywords: Hopf bifurcation, singular perturbation, Liénard equation., Slow-fast system, canard cycle, turning point.
Mathematics Subject Classification: 34C05, 34C07, 34C08, 34C23, 34C25, 34C26, 34E15, 34E2.
Citation: Freddy Dumortier, Robert Roussarie. Birth of canard cycles. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 723-781. doi: 10.3934/dcdss.2009.2.723
[1]

Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021097
[2]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[3]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[4]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026
[5]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[6]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098
[7]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[8]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031
[9]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073
[10]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047
[11]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032
[12]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041
[13]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009
[14]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[15]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403
[16]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037
[17]

Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012
[18]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025
[19]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027
[20]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences