American Institute of Mathematical Sciences

December  2007, 19(4): 631-674. doi: 10.3934/dcds.2007.19.631

Bifurcation of relaxation oscillations in dimension two

 1 Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium 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  December 2006 Revised  June 2007 Published  September 2007

The paper deals with the bifurcation of relaxation oscillations in two dimensional slow-fast systems. The most generic case is studied by means of geometric singular perturbation theory, using blow up at contact points. It reveals that the bifurcation goes through a continuum of transient canard oscillations, controlled by the slow divergence integral along the critical curve. The theory is applied to polynomial Liénard equations, showing that the cyclicity near a generic coallescence of two relaxation oscillations does not need to be limited to two, but can be arbitrarily high.
Citation: Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631
 [1] 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 [2] Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289 [3] Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2641-2673. doi: 10.3934/cpaa.2014.13.2641 [4] Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021 [5] 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 [6] Hong Li. Bifurcation of limit cycles from a Li$\acute{E}$nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022033 [7] Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233 [8] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 [9] Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171 [10] Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112 [11] Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085 [12] Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237 [13] Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187 [14] Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043 [15] Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621 [16] Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6285-6310. doi: 10.3934/dcdsb.2021019 [17] Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1447-1469. doi: 10.3934/dcdsb.2021097 [18] Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127 [19] Liang Zhao, Jianhe Shen. Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022018 [20] A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126

2020 Impact Factor: 1.392