- Previous Article
- CPAA Home
- This Issue
-
Next Article
Positive solutions for Robin problems with general potential and logistic reaction
On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line
1. | School of Science, Jiangnan University, Wuxi, 214122, China |
2. | Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795 |
3. | Institute for Intelligent Systems, the University of Johannesburg, South Africa |
4. | Department of Mathematics, Tongji University, Shanghai, 200092, China |
References:
[1] |
Y. L. An and M. A. Han, On the number of limit cycles near a homoclinic loop with a nilpotent singular point,, \emph{J. Differential Equations}, 258 (2015), 3194.
doi: 10.1016/j.jde.2015.01.006. |
[2] |
J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles,, \emph{Math. Comput. Simulation}, 95 (2014), 13.
doi: 10.1016/j.matcom.2013.02.007. |
[3] |
V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems,, \emph{Phys. D}, 248 (2013), 44.
doi: 10.1016/j.physd.2013.01.002. |
[4] |
Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems,, \emph{Nonlinear Anal.}, 69 (2008), 3610.
doi: 10.1016/j.na.2007.09.045. |
[5] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series),, Kluwer Academic Publishers Group, (1988).
doi: 10.1007/978-94-015-7793-9. |
[6] |
N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model,, \emph{J. Dynam. Differential Equations}, 26 (2014), 1133.
doi: 10.1007/s10884-014-9403-0. |
[7] |
E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, \emph{SIAM J. Appl. Dyn. Syst.}, 11 (2012), 181.
doi: 10.1137/11083928X. |
[8] |
Z. Y. Hou and S. Baigent, Global stability and repulsion in autonomous kolmogorov systems,, \emph{Commun. Pure Appl. Anal}, 14 ().
doi: 10.3934/cpaa.2015.14.1205. |
[9] |
S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics,, \emph{Nonlinear Anal.}, 92 (2013), 82.
doi: 10.1016/j.na.2013.06.017. |
[10] |
S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types,, \emph{J. Math. Anal. Appl.}, 411 (2014), 340.
doi: 10.1016/j.jmaa.2013.08.064. |
[11] |
T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions,, \emph{J. Phys. A}, 38 (2005), 8211.
doi: 10.1088/0305-4470/38/38/003. |
[12] |
I. D. Iliev, C. Z. Li and J. Yu, Bifurcations of limit cycles in reversible quadratic system with a center, a saddle and two nodes,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 583.
doi: 10.3934/cpaa.2010.9.583. |
[13] |
F. Jiang and J. Sun, On the uniqueness of limit cycles in discontinuous Liénard-type systems,, \emph{Electron. J. Qual. Theory Differ. Equ.}, (): 1.
|
[14] |
F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems,, \emph{Int. J. Bifurcat. Chaos}, 25 (2015).
doi: 10.1142/S021812741550131X. |
[15] |
F. Jiang and J. Sun, Existence and uniqueness of limit cycle in discontinuous planar differential systems,, \emph{Qual. Theor. Dyn. Syst.}, (): 1.
doi: 10.1007/s12346-015-0141-4. |
[16] |
Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.
doi: 10.1016/0025-5564(88)90049-1. |
[17] |
C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four,, \emph{J. Differential Equations}, 252 (2012), 3142.
doi: 10.1016/j.jde.2011.11.002. |
[18] |
P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model,, \emph{J. Math. Chem.}, 51 (2013), 2001.
doi: 10.1007/s10910-013-0196-x. |
[19] |
J. Llibre and A. C. Mereu, Limit cycles for discontinuous generalized Liénard polynomial differential equations,, \emph{Electron. J. Differential Equations}, ().
|
[20] |
J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree,, \emph{Nonlinearity}, 27 (2014), 563.
doi: 10.1088/0951-7715/27/3/563. |
[21] |
J. Llibre, M. Ordó nez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 2002.
doi: 10.1016/j.nonrwa.2013.02.004. |
[22] |
J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities,, \emph{Nonlinearity}, 21 (2008), 2121.
doi: 10.1088/0951-7715/21/9/013. |
[23] |
A. C. J. Luo, Discontinuous Dynamical Systems,, Higher Education Press, (2012).
doi: 10.1007/978-3-642-22461-4. |
[24] |
J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, \emph{J. Theoret. Biol.}, 81 (1979), 389.
doi: 10.1016/0022-5193(79)90042-0. |
[25] |
J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, \emph{J. Math. Biol.}, 62 (2011), 291.
doi: 10.1007/s00285-010-0332-1. |
[26] |
D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems,, \emph{Nonlinearity}, 16 (2003), 1185.
doi: 10.1088/0951-7715/16/3/321. |
[27] |
Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M.-S. Wang and X.-A. Yang, Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs,, 2nd edition, ().
|
[28] |
Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations,, \emph{Appl. Anal.}, 23 (1986), 63.
doi: 10.1080/00036818608839631. |
[29] |
Z. F. Zhang, T. R. Ding, W. Z. Huang and Z.-X. Dong, Qualitative theory of differential equations, vol. 101 of Translations of Mathematical Monographs,, American Mathematical Society, (1992). Google Scholar |
show all references
References:
[1] |
Y. L. An and M. A. Han, On the number of limit cycles near a homoclinic loop with a nilpotent singular point,, \emph{J. Differential Equations}, 258 (2015), 3194.
doi: 10.1016/j.jde.2015.01.006. |
[2] |
J. C. Artés, J. Llibre, J. C. Medrado and M. A. Teixeira, Piecewise linear differential systems with two real saddles,, \emph{Math. Comput. Simulation}, 95 (2014), 13.
doi: 10.1016/j.matcom.2013.02.007. |
[3] |
V. Carmona, S. Fernández-García, E. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems,, \emph{Phys. D}, 248 (2013), 44.
doi: 10.1016/j.physd.2013.01.002. |
[4] |
Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems,, \emph{Nonlinear Anal.}, 69 (2008), 3610.
doi: 10.1016/j.na.2007.09.045. |
[5] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series),, Kluwer Academic Publishers Group, (1988).
doi: 10.1007/978-94-015-7793-9. |
[6] |
N. Forcadel, A. Ghorbel and S. Walha, Existence and uniqueness of traveling wave for accelerated Frenkel-Kontorova model,, \emph{J. Dynam. Differential Equations}, 26 (2014), 1133.
doi: 10.1007/s10884-014-9403-0. |
[7] |
E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems,, \emph{SIAM J. Appl. Dyn. Syst.}, 11 (2012), 181.
doi: 10.1137/11083928X. |
[8] |
Z. Y. Hou and S. Baigent, Global stability and repulsion in autonomous kolmogorov systems,, \emph{Commun. Pure Appl. Anal}, 14 ().
doi: 10.3934/cpaa.2015.14.1205. |
[9] |
S. Huan and X. Yang, Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics,, \emph{Nonlinear Anal.}, 92 (2013), 82.
doi: 10.1016/j.na.2013.06.017. |
[10] |
S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types,, \emph{J. Math. Anal. Appl.}, 411 (2014), 340.
doi: 10.1016/j.jmaa.2013.08.064. |
[11] |
T. W. Hwang and H. J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions,, \emph{J. Phys. A}, 38 (2005), 8211.
doi: 10.1088/0305-4470/38/38/003. |
[12] |
I. D. Iliev, C. Z. Li and J. Yu, Bifurcations of limit cycles in reversible quadratic system with a center, a saddle and two nodes,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 583.
doi: 10.3934/cpaa.2010.9.583. |
[13] |
F. Jiang and J. Sun, On the uniqueness of limit cycles in discontinuous Liénard-type systems,, \emph{Electron. J. Qual. Theory Differ. Equ.}, (): 1.
|
[14] |
F. Jiang, J. Shi and J. Sun, On the number of limit cycles for discontinuous generalized linéard polynomial differential systems,, \emph{Int. J. Bifurcat. Chaos}, 25 (2015).
doi: 10.1142/S021812741550131X. |
[15] |
F. Jiang and J. Sun, Existence and uniqueness of limit cycle in discontinuous planar differential systems,, \emph{Qual. Theor. Dyn. Syst.}, (): 1.
doi: 10.1007/s12346-015-0141-4. |
[16] |
Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems,, \emph{Math. Biosci.}, 88 (1988), 67.
doi: 10.1016/0025-5564(88)90049-1. |
[17] |
C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four,, \emph{J. Differential Equations}, 252 (2012), 3142.
doi: 10.1016/j.jde.2011.11.002. |
[18] |
P. Liu, J. P. Shi, Y. W. Wang and X. H. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model,, \emph{J. Math. Chem.}, 51 (2013), 2001.
doi: 10.1007/s10910-013-0196-x. |
[19] |
J. Llibre and A. C. Mereu, Limit cycles for discontinuous generalized Liénard polynomial differential equations,, \emph{Electron. J. Differential Equations}, ().
|
[20] |
J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree,, \emph{Nonlinearity}, 27 (2014), 563.
doi: 10.1088/0951-7715/27/3/563. |
[21] |
J. Llibre, M. Ordó nez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry,, \emph{Nonlinear Anal. Real World Appl.}, 14 (2013), 2002.
doi: 10.1016/j.nonrwa.2013.02.004. |
[22] |
J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities,, \emph{Nonlinearity}, 21 (2008), 2121.
doi: 10.1088/0951-7715/21/9/013. |
[23] |
A. C. J. Luo, Discontinuous Dynamical Systems,, Higher Education Press, (2012).
doi: 10.1007/978-3-642-22461-4. |
[24] |
J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, \emph{J. Theoret. Biol.}, 81 (1979), 389.
doi: 10.1016/0022-5193(79)90042-0. |
[25] |
J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, \emph{J. Math. Biol.}, 62 (2011), 291.
doi: 10.1007/s00285-010-0332-1. |
[26] |
D. M. Xiao and Z. F. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems,, \emph{Nonlinearity}, 16 (2003), 1185.
doi: 10.1088/0951-7715/16/3/321. |
[27] |
Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M.-S. Wang and X.-A. Yang, Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs,, 2nd edition, ().
|
[28] |
Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations,, \emph{Appl. Anal.}, 23 (1986), 63.
doi: 10.1080/00036818608839631. |
[29] |
Z. F. Zhang, T. R. Ding, W. Z. Huang and Z.-X. Dong, Qualitative theory of differential equations, vol. 101 of Translations of Mathematical Monographs,, American Mathematical Society, (1992). Google Scholar |
[1] |
Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 |
[2] |
Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326 |
[3] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[4] |
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 |
[5] |
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257 |
[6] |
Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145 |
[7] |
Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021002 |
[8] |
Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020368 |
[9] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[10] |
Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020405 |
[11] |
Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020320 |
[12] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[13] |
Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 |
[14] |
Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020353 |
[15] |
Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 |
[16] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289 |
[17] |
Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 |
[18] |
Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 |
[19] |
Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030 |
[20] |
Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]