Prevalence of stable periodic solutions in the forced relativistic pendulum equation

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December 2018, 23(10): 4579-4594. doi: 10.3934/dcdsb.2018177

Prevalence of stable periodic solutions in the forced relativistic pendulum equation

Feng Wang ^1,2, , Jifeng Chu ^3,, and Zaitao Liang ^4,

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

* Corresponding author: Jifeng Chu

Received September 2017 Revised January 2018 Published June 2018

Fund Project: Feng Wang was sponsored by Qing Lan Project of Jiangsu Province, and was supported by the National Natural Science Foundation of China (Grant No. 11501055 and No. 11401166), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110001), China Postdoctoral Science Foundation funded project (Grant No. 2017M610315), Hainan Natural Science Foundation (Grant No.117005). Jifeng Chu was supported by the National Natural Science Foundation of China (Grant No. 11671118)

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.

Keywords: Prevalence, stable periodic solutions, forced relativistic pendulum equation, planar system.

Mathematics Subject Classification: Primary: 34C25.

Citation: Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177

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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