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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  December 2018 Early access  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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Magnus and S. Winkler, Hill Equations, corrected reprint of 1966 edition, Dover, New York, 1979.  Google Scholar

[13]

S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.   Google Scholar

[14]

J. Mawhin, Global results for the forced pendulum equations, in: Handbook of Differential Equations, Ordinary Differential Equations, Elsevier, Amsterdam, 1 (2004), 533-589.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. Ortega, Stable periodic solutions in the forced pendulum equation, Regul. Chaotic Dyn., 18 (2013), 585-599.  doi: 10.1134/S1560354713060026.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Ott and J. A. Yorke, Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.  doi: 10.1090/S0273-0979-05-01060-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801-810.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Magnus and S. Winkler, Hill Equations, corrected reprint of 1966 edition, Dover, New York, 1979.  Google Scholar

[13]

S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics, Topol. Methods Nonlinear Anal., 42 (2013), 51-75.   Google Scholar

[14]

J. Mawhin, Global results for the forced pendulum equations, in: Handbook of Differential Equations, Ordinary Differential Equations, Elsevier, Amsterdam, 1 (2004), 533-589.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. Ortega, Stable periodic solutions in the forced pendulum equation, Regul. Chaotic Dyn., 18 (2013), 585-599.  doi: 10.1134/S1560354713060026.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

W. Ott and J. A. Yorke, Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.  doi: 10.1090/S0273-0979-05-01060-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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