American Institute of Mathematical Sciences

Advanced
April  2007, 17(2): 441-448. doi: 10.3934/dcds.2007.17.441

Putting a boundary to the space of Liénard equations

Robert Roussarie 1,

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

Received  December 2005 Revised  September 2006 Published  November 2006

A lot of partial results are known about the Liénard differential equations : $\dot x= y -F_a^n(x),\ \ \dot y =-x.$ Here $F_a^n$ is a polynomial of degree $2n+1,\ \ F_a^n(x)= \sum_{i=1}^{2n}a_ix^i+x^{2n+1},$ where $a = (a_1,\cdots,a_{2n}) \in \R^{2n}.$ For instance, it is easy to see that for any $a$ the related vector field $X_a$ has just a finite number of limit cycles. This comes from the fact that $X_a$ has a global return map on the half-axis $Ox=\{x \geq 0\},$ and that this map is analytic and repelling at infinity. It is also easy to verify that at most $n$ limit cycles can bifurcate from the origin. For these reasons, Lins Neto, de Melo and Pugh have conjectured that the total number of limit cycles is also bounded by $n,$ in the whole plane and for any value $a.$
    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 slow-fast 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 slow-fast cycles of these singular equations.
Keywords: limit cycles, Liénard equation, Cyclicity, slow-fast equation..
Mathematics Subject Classification: Primary: 34C05; Secondary: 34C2.
Citation: Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[1]

Renato Huzak. Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 171-215. 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) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[3]

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
[4]

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
[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) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[6]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. 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) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129
[8]

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
[9]

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
[10]

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
[11]

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
[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) : 2465-2478. 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) : 25-34. doi: 10.3934/dcds.2011.31.25
[14]

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
[15]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. 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 slow-fast systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[17]

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
[18]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019
[19]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. 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) : 449-461. doi: 10.3934/eect.2016013

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences