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Prevalence of stable periodic solutions in the forced relativistic pendulum equation
1. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
2. | School of Mathematics and Physics, Changzhou University, Changzhou 213164, China |
3. | Department of Mathematics, Shanghai Normal University, Shanghai 200234, China |
4. | College of Science, Hohai University, Nanjing 210098, China |
We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.
References:
[1] |
C. Bereanu, P. Jebelean and J. Mawhin,
Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463-471.
doi: 10.1007/s10884-010-9172-3. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin,
Multiple solutions for Neumann and periodic problems with singular $\phi$-Laplacian, J. Funct. Anal., 261 (2011), 3226-3246.
doi: 10.1016/j.jfa.2011.07.027. |
[3] |
C. Bereanu and P. J. Torres,
Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2719.
doi: 10.1090/S0002-9939-2011-11101-8. |
[4] |
H. Brezis and J. Mawhin,
Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.
|
[5] |
J. Chu, J. Lei and M. Zhang,
The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.
doi: 10.1016/j.jde.2008.11.013. |
[6] |
J. Chu, Z. Liang, F. Liao and S. Lu,
Existence and stability of periodic solutions for relativistic singular equations, Commun. Pure Appl. Anal., 16 (2017), 591-609.
doi: 10.3934/cpaa.2017029. |
[7] |
J. Chu and F. Wang,
Prevalence of stable periodic solutions for Duffing equations, J. Differential Equations, 260 (2016), 7800-7820.
doi: 10.1016/j.jde.2016.02.003. |
[8] |
J. A. Cid and P. J. Torres,
On the existence and stability of periodic solutions for pendulum-like equations with friction and ϕ-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141-153.
doi: 10.3934/dcds.2013.33.141. |
[9] |
Y. Hua and Z. Xia,
Stability of elliptic periodic points with an application to Lagrangian equilibrium solutions, Qual. Theory Dyn. Syst., 12 (2013), 243-253.
doi: 10.1007/s12346-012-0093-x. |
[10] |
P. Jebelean, J. Mawhin and C. Serban,
Multiple periodic solutions for perturbed relativistic pendulum systems, Proc. Amer. Math. Soc., 143 (2015), 3029-3039.
doi: 10.1090/S0002-9939-2015-12542-7. |
[11] |
B. M. Levitan and L. S. Sargsjan, Sturm-Liouville and Dirac Operators, Math. Appl. (Soviet Ser.), vol. 59, Kluwer Academic, Dordrecht, 1991.
doi: 10.1007/978-94-011-3748-5. |
[12] |
W. Magnus and S. Winkler, Hill Equations, corrected reprint of 1966 edition, Dover, New York, 1979. |
[13] |
S. Marò,
Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.
|
[14] |
J. Mawhin, Global results for the forced pendulum equations, in: Handbook of Differential Equations, Ordinary Differential Equations, Elsevier, Amsterdam, 1 (2004), 533-589. |
[15] |
R. Ortega, Some applications of the topological degree to stability theory, in: Topological Methods in Differential Equations and Inclusions, Kluwer Acad. Publ., Dordrecht, 472 (1995), 377-409. |
[16] |
R. Ortega,
Prevalence of non-degenerate periodic solutions in the forced pendulum equation, Adv. Nonlinear Stud., 13 (2013), 219-229.
doi: 10.1515/ans-2013-0113. |
[17] |
R. Ortega,
Stable periodic solutions in the forced pendulum equation, Regul. Chaotic Dyn., 18 (2013), 585-599.
doi: 10.1134/S1560354713060026. |
[18] |
R. Ortega,
A forced pendulum equation without stable periodic solutions of a fixed period, Port. Math., 7 (2014), 193-216.
doi: 10.4171/PM/1950. |
[19] |
R. Ortega and M. Zhang,
Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.
doi: 10.1017/S0308210500003796. |
[20] |
W. Ott and J. A. Yorke,
Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.
doi: 10.1090/S0273-0979-05-01060-8. |
[21] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[22] |
P. J. Torres,
Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386-6387.
doi: 10.1016/j.physleta.2008.08.060. |
show all references
References:
[1] |
C. Bereanu, P. Jebelean and J. Mawhin,
Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463-471.
doi: 10.1007/s10884-010-9172-3. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin,
Multiple solutions for Neumann and periodic problems with singular $\phi$-Laplacian, J. Funct. Anal., 261 (2011), 3226-3246.
doi: 10.1016/j.jfa.2011.07.027. |
[3] |
C. Bereanu and P. J. Torres,
Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713-2719.
doi: 10.1090/S0002-9939-2011-11101-8. |
[4] |
H. Brezis and J. Mawhin,
Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.
|
[5] |
J. Chu, J. Lei and M. Zhang,
The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.
doi: 10.1016/j.jde.2008.11.013. |
[6] |
J. Chu, Z. Liang, F. Liao and S. Lu,
Existence and stability of periodic solutions for relativistic singular equations, Commun. Pure Appl. Anal., 16 (2017), 591-609.
doi: 10.3934/cpaa.2017029. |
[7] |
J. Chu and F. Wang,
Prevalence of stable periodic solutions for Duffing equations, J. Differential Equations, 260 (2016), 7800-7820.
doi: 10.1016/j.jde.2016.02.003. |
[8] |
J. A. Cid and P. J. Torres,
On the existence and stability of periodic solutions for pendulum-like equations with friction and ϕ-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141-153.
doi: 10.3934/dcds.2013.33.141. |
[9] |
Y. Hua and Z. Xia,
Stability of elliptic periodic points with an application to Lagrangian equilibrium solutions, Qual. Theory Dyn. Syst., 12 (2013), 243-253.
doi: 10.1007/s12346-012-0093-x. |
[10] |
P. Jebelean, J. Mawhin and C. Serban,
Multiple periodic solutions for perturbed relativistic pendulum systems, Proc. Amer. Math. Soc., 143 (2015), 3029-3039.
doi: 10.1090/S0002-9939-2015-12542-7. |
[11] |
B. M. Levitan and L. S. Sargsjan, Sturm-Liouville and Dirac Operators, Math. Appl. (Soviet Ser.), vol. 59, Kluwer Academic, Dordrecht, 1991.
doi: 10.1007/978-94-011-3748-5. |
[12] |
W. Magnus and S. Winkler, Hill Equations, corrected reprint of 1966 edition, Dover, New York, 1979. |
[13] |
S. Marò,
Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.
|
[14] |
J. Mawhin, Global results for the forced pendulum equations, in: Handbook of Differential Equations, Ordinary Differential Equations, Elsevier, Amsterdam, 1 (2004), 533-589. |
[15] |
R. Ortega, Some applications of the topological degree to stability theory, in: Topological Methods in Differential Equations and Inclusions, Kluwer Acad. Publ., Dordrecht, 472 (1995), 377-409. |
[16] |
R. Ortega,
Prevalence of non-degenerate periodic solutions in the forced pendulum equation, Adv. Nonlinear Stud., 13 (2013), 219-229.
doi: 10.1515/ans-2013-0113. |
[17] |
R. Ortega,
Stable periodic solutions in the forced pendulum equation, Regul. Chaotic Dyn., 18 (2013), 585-599.
doi: 10.1134/S1560354713060026. |
[18] |
R. Ortega,
A forced pendulum equation without stable periodic solutions of a fixed period, Port. Math., 7 (2014), 193-216.
doi: 10.4171/PM/1950. |
[19] |
R. Ortega and M. Zhang,
Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.
doi: 10.1017/S0308210500003796. |
[20] |
W. Ott and J. A. Yorke,
Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.
doi: 10.1090/S0273-0979-05-01060-8. |
[21] |
A. Szulkin,
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 77-109.
doi: 10.1016/S0294-1449(16)30389-4. |
[22] |
P. J. Torres,
Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386-6387.
doi: 10.1016/j.physleta.2008.08.060. |
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