Bifurcation of relaxation oscillations in dimension two

Previous Article
Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals
DCDS Home
This Issue
Next Article
Action functionals that attain regular minima in presence of energy gaps

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

Bifurcation of relaxation oscillations in dimension two

Freddy Dumortier ^1, and Robert Roussarie ^2,

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

Abstract
Related Papers

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.

Keywords: canard cycle, Slow-fast system, Liénard equation., relaxation oscillation, bifurcation.

Mathematics Subject Classification: Primary: 34C05, 34C07, 34C23, 34C26, 34E1.

Citation: Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893
[6]	Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233
[7]	Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[8]	Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 171-215. doi: 10.3934/dcds.2016.36.171
[9]	Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[10]	Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[11]	Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[12]	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 & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[13]	Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[14]	Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[15]	A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[16]	Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
[17]	Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[18]	Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[19]	Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[20]	Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences