 Previous Article
 DCDS Home
 This Issue

Next Article
On the Burnside problem in Diff(M)
Putting a boundary to the space of Liénard equations
1.  Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France 
In fact it is not even known if there exists a finite bound $L(n)$ independent of $a,$ for the number of limit cycles. In this paper, I want to investigate this question of finiteness. I show that there exists a finite bound $L(K,n)$ if one restricts the parameter in a compact $K$ and that there is a natural way to put a boundary to the space of Liénard equations. This boundary is made of slowfast equations of Liénard type, obtained as singular limits of the Liénard equations for large values of the parameter. Then the existence of a global bound $L(n)$ can be related to the finiteness of the number of limit cycles which bifurcate from slowfast cycles of these singular equations.
[1] 
Renato Huzak. Cyclicity of the origin in slowfast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171215. doi: 10.3934/dcds.2016.36.171 
[2] 
Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 21272150. doi: 10.3934/cpaa.2015.14.2127 
[3] 
Ilya Schurov. Duck farming on the twotorus: Multiple canard cycles in generic slowfast systems. Conference Publications, 2011, 2011 (Special) : 12891298. doi: 10.3934/proc.2011.2011.1289 
[4] 
Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slowfast codimension 3 elliptic bifurcations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 26412673. doi: 10.3934/cpaa.2014.13.2641 
[5] 
Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 30853108. doi: 10.3934/dcds.2013.33.3085 
[6] 
Min Hu, Tao Li, Xingwu Chen. Bicenter problem and Hopf cyclicity of a Cubic Liénard system. Discrete & Continuous Dynamical Systems  B, 2020, 25 (1) : 401414. doi: 10.3934/dcdsb.2019187 
[7] 
Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems  B, 2014, 19 (4) : 11291136. doi: 10.3934/dcdsb.2014.19.1129 
[8] 
Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slowfast SPDEs with Poisson random measures. Discrete & Continuous Dynamical Systems  B, 2015, 20 (7) : 22332256. doi: 10.3934/dcdsb.2015.20.2233 
[9] 
Alexandre Vidal. Periodic orbits of tritrophic slowfast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 10211030. doi: 10.3934/proc.2007.2007.1021 
[10] 
Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slowfast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems  B, 2015, 20 (7) : 22572267. doi: 10.3934/dcdsb.2015.20.2257 
[11] 
Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slowfast dynamics. Discrete & Continuous Dynamical Systems  B, 2018, 23 (7) : 29352950. doi: 10.3934/dcdsb.2018112 
[12] 
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) : 24652478. doi: 10.3934/dcdsb.2017126 
[13] 
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 2534. doi: 10.3934/dcds.2011.31.25 
[14] 
Fangfang Jiang, Junping Shi, Qingguo 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) : 25092526. doi: 10.3934/cpaa.2016047 
[15] 
Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 10431057. doi: 10.3934/dcds.2002.8.1043 
[16] 
Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slowfast systems. Discrete & Continuous Dynamical Systems  B, 2008, 10 (2&3, September) : 621650. doi: 10.3934/dcdsb.2008.10.621 
[17] 
Chunhua Shan. Slowfast dynamics and nonlinear oscillations in transmission of mosquitoborne diseases. Discrete & Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021097 
[18] 
Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slowfast stochastic reactiondiffusion equations. Discrete & Continuous Dynamical Systems  B, 2021 doi: 10.3934/dcdsb.2021019 
[19] 
Andrea Giorgini. On the SwiftHohenberg equation with slow and fast dynamics: wellposedness and longtime behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219241. doi: 10.3934/cpaa.2016.15.219 
[20] 
Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449461. doi: 10.3934/eect.2016013 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]