
Previous Article
Boundary stabilization of the waves in partially rectangular domains
 DCDS Home
 This Issue

Next Article
SRB attractors with intermingled basins for nonhyperbolic diffeomorphisms
Periodic solutions of Liénard equations with resonant isochronous potentials
1.  School of Mathematical Sciences, Capital Normal University, Beijing 100048, China, China 
References:
[1] 
J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201220. doi: 10.1006/jdeq.1997.3367. 
[2] 
D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907930. 
[3] 
D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral equations, 15 (2002), 11391152. 
[4] 
A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119132. doi: 10.1112/S0024610703004459. 
[5] 
A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance, Proc. Edinburgh Math. Soc., 138A (2008), 1532. 
[6] 
A. Capietto, W. Dambrosio, T. Ma and Z. Wang, Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance,, Discrete Contin. Dynam. Systems, (). 
[7] 
N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321328. doi: 10.1017/S0004972700022747. 
[8] 
P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator, J. Differential Equations, 5 (1969), 262270. 
[9] 
T. Ma and Z. Wang, Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials, Z. Angew. Math. Phys., 63 (2012), 2549. doi: 10.1007/s0003301101521. 
[10] 
T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Springer, New York, 1975. 
[11] 
C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 5878. doi: 10.1006/jdeq.1998.3441. 
[12] 
C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations, 214 (2005), 299325. doi: 10.1016/j.jde.2005.02.003. 
[13] 
C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493505. doi: 10.1088/09517715/13/3/302. 
[14] 
A. Fonda and M. Garrione, Double resonance with LandesmanLazer conditions for planar systems of ordinary differential equations, J. Differential Equations, 250 (2011), 10521082. doi: 10.1016/j.jde.2010.08.006. 
[15] 
A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Diff. Eqns, 11 (2006), 11111133. 
[16] 
S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems," Reidel, Boston, 1980. 
[17] 
A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura. Appl., 82 (1969), 4968. doi: 10.1007/BF02410787. 
[18] 
A. C. Lazer and P. J. McKenna, Existence, uniqueness and stability of oscillators in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721739. doi: 10.1090/S00029947198909799631. 
[19] 
J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics," Vol. 1, Oxford: Pergamon, 1960. 
[20] 
Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751770. doi: 10.3934/dcds.2003.9.751. 
[21] 
D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the LazerLeachDancer condition, J. Differential Equations, 171 (2001), 233250. doi: 10.1006/jdeq.2000.3847. 
[22] 
M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," SpringerVerlag, 1984. 
show all references
References:
[1] 
J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201220. doi: 10.1006/jdeq.1997.3367. 
[2] 
D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907930. 
[3] 
D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral equations, 15 (2002), 11391152. 
[4] 
A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119132. doi: 10.1112/S0024610703004459. 
[5] 
A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance, Proc. Edinburgh Math. Soc., 138A (2008), 1532. 
[6] 
A. Capietto, W. Dambrosio, T. Ma and Z. Wang, Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance,, Discrete Contin. Dynam. Systems, (). 
[7] 
N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321328. doi: 10.1017/S0004972700022747. 
[8] 
P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator, J. Differential Equations, 5 (1969), 262270. 
[9] 
T. Ma and Z. Wang, Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials, Z. Angew. Math. Phys., 63 (2012), 2549. doi: 10.1007/s0003301101521. 
[10] 
T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Springer, New York, 1975. 
[11] 
C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 5878. doi: 10.1006/jdeq.1998.3441. 
[12] 
C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations, 214 (2005), 299325. doi: 10.1016/j.jde.2005.02.003. 
[13] 
C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493505. doi: 10.1088/09517715/13/3/302. 
[14] 
A. Fonda and M. Garrione, Double resonance with LandesmanLazer conditions for planar systems of ordinary differential equations, J. Differential Equations, 250 (2011), 10521082. doi: 10.1016/j.jde.2010.08.006. 
[15] 
A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Diff. Eqns, 11 (2006), 11111133. 
[16] 
S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems," Reidel, Boston, 1980. 
[17] 
A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura. Appl., 82 (1969), 4968. doi: 10.1007/BF02410787. 
[18] 
A. C. Lazer and P. J. McKenna, Existence, uniqueness and stability of oscillators in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721739. doi: 10.1090/S00029947198909799631. 
[19] 
J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics," Vol. 1, Oxford: Pergamon, 1960. 
[20] 
Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751770. doi: 10.3934/dcds.2003.9.751. 
[21] 
D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the LazerLeachDancer condition, J. Differential Equations, 171 (2001), 233250. doi: 10.1006/jdeq.2000.3847. 
[22] 
M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," SpringerVerlag, 1984. 
[1] 
A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete and Continuous Dynamical Systems  B, 2017, 22 (6) : 24652478. doi: 10.3934/dcdsb.2017126 
[2] 
Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems  B, 2011, 16 (3) : 703717. doi: 10.3934/dcdsb.2011.16.703 
[3] 
Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$Laplacian systems of Liénardtype. Communications on Pure and Applied Analysis, 2011, 10 (5) : 13931400. doi: 10.3934/cpaa.2011.10.1393 
[4] 
Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 10431057. doi: 10.3934/dcds.2002.8.1043 
[5] 
Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 21272150. doi: 10.3934/cpaa.2015.14.2127 
[6] 
Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete and Continuous Dynamical Systems  B, 2016, 21 (2) : 557573. doi: 10.3934/dcdsb.2016.21.557 
[7] 
Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 137160. doi: 10.3934/dcds.2005.12.137 
[8] 
Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 441448. doi: 10.3934/dcds.2007.17.441 
[9] 
Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete and Continuous Dynamical Systems  B, 2008, 10 (2&3, September) : 485494. doi: 10.3934/dcdsb.2008.10.485 
[10] 
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 and Applied Analysis, 2016, 15 (6) : 25092526. doi: 10.3934/cpaa.2016047 
[11] 
Jitsuro Sugie, Tadayuki Hara. Existence and nonexistence of homoclinic trajectories of the Liénard system. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 237254. doi: 10.3934/dcds.1996.2.237 
[12] 
Min Hu, Tao Li, Xingwu Chen. Bicenter problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems  B, 2020, 25 (1) : 401414. doi: 10.3934/dcdsb.2019187 
[13] 
D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 907930. doi: 10.3934/dcds.2002.8.907 
[14] 
Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 33473371. doi: 10.3934/cpaa.2021108 
[15] 
Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 30853108. doi: 10.3934/dcds.2013.33.3085 
[16] 
Hong Li. Bifurcation of limit cycles from a Li$ \acute{E} $nard system with asymmetric figure eightloop case. Discrete and Continuous Dynamical Systems  S, 2022 doi: 10.3934/dcdss.2022033 
[17] 
Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 18351856. doi: 10.3934/dcds.2013.33.1835 
[18] 
Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$concentration of the blowup solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119127. doi: 10.3934/mcrf.2011.1.119 
[19] 
Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive GrossPitaevskii equation with periodic potential: Coding and method for computation. Discrete and Continuous Dynamical Systems  B, 2017, 22 (4) : 12071229. doi: 10.3934/dcdsb.2017059 
[20] 
Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized LandauLifshitzBloch equation in high dimensions. Discrete and Continuous Dynamical Systems  B, 2020, 25 (4) : 13451360. doi: 10.3934/dcdsb.2019230 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]