American Institute of Mathematical Sciences

Advanced
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.  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

[1]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139
[2]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[3]

Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
[4]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[5]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029
[6]

D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907
[7]

Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103
[8]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[9]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341
[10]

Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations & Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135
[11]

Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393
[12]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[13]

J. Ángel Cid, Pedro J. Torres. On the existence and stability of periodic solutions for pendulum-like equations with friction and $\phi$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 141-152. doi: 10.3934/dcds.2013.33.141
[14]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373
[15]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[16]

Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
[17]

Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024
[18]

E. N. Dancer, Norimichi Hirano. Existence of stable and unstable periodic solutions for semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 207-216. doi: 10.3934/dcds.1997.3.207
[19]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157
[20]

Stephen Pankavich, Nicholas Michalowski. Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system. Kinetic & Related Models, 2015, 8 (1) : 169-199. doi: 10.3934/krm.2015.8.169

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (72)
  • HTML views (298)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences