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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, J. Differential Equations, 258 (2015), 3194-3247.
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, Math. Comput. Simulation, 95 (2014), 13-22.
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, Phys. D, 248 (2013), 44-54.
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, Nonlinear Anal., 69 (2008), 3610-3628.
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, Dordrecht, 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, J. Dynam. Differential Equations, 26 (2014), 1133-1169.
doi: 10.1007/s10884-014-9403-0. |
[7] |
E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.
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, Nonlinear Anal., 92 (2013), 82-95.
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, J. Math. Anal. Appl., 411 (2014), 340-353.
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, J. Phys. A, 38 (2005), 8211-8223.
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, Commun. Pure Appl. Anal, 9 (2010), 583-610.
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, Int. J. Bifurcat. Chaos, 25 (2015), 1550131.
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, Math. Biosci., 88 (1988), 67-84.
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, J. Differential Equations, 252 (2012), 3142-3162.
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, J. Math. Chem., 51 (2013), 2001-2019.
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, Nonlinearity, 27 (2014), 563-583.
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, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012.
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, Nonlinearity, 21 (2008), 2121-2142.
doi: 10.1088/0951-7715/21/9/013. |
[23] |
A. C. J. Luo, Discontinuous Dynamical Systems, Higher Education Press, Beijing; Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-22461-4. |
[24] |
J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400.
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, J. Math. Biol., 62 (2011), 291-331.
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, Nonlinearity, 16 (2003), 1185-1201.
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, Appl. Anal., 23 (1986), 63-76.
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, Providence, RI, 1992, |
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, J. Differential Equations, 258 (2015), 3194-3247.
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, Math. Comput. Simulation, 95 (2014), 13-22.
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, Phys. D, 248 (2013), 44-54.
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, Nonlinear Anal., 69 (2008), 3610-3628.
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, Dordrecht, 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, J. Dynam. Differential Equations, 26 (2014), 1133-1169.
doi: 10.1007/s10884-014-9403-0. |
[7] |
E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211.
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, Nonlinear Anal., 92 (2013), 82-95.
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, J. Math. Anal. Appl., 411 (2014), 340-353.
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, J. Phys. A, 38 (2005), 8211-8223.
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, Commun. Pure Appl. Anal, 9 (2010), 583-610.
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, Int. J. Bifurcat. Chaos, 25 (2015), 1550131.
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, Math. Biosci., 88 (1988), 67-84.
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, J. Differential Equations, 252 (2012), 3142-3162.
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, J. Math. Chem., 51 (2013), 2001-2019.
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, Nonlinearity, 27 (2014), 563-583.
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, Nonlinear Anal. Real World Appl., 14 (2013), 2002-2012.
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, Nonlinearity, 21 (2008), 2121-2142.
doi: 10.1088/0951-7715/21/9/013. |
[23] |
A. C. J. Luo, Discontinuous Dynamical Systems, Higher Education Press, Beijing; Springer, Heidelberg, 2012.
doi: 10.1007/978-3-642-22461-4. |
[24] |
J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81 (1979), 389-400.
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, J. Math. Biol., 62 (2011), 291-331.
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, Nonlinearity, 16 (2003), 1185-1201.
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, Appl. Anal., 23 (1986), 63-76.
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, Providence, RI, 1992, |
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