American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and uniqueness of solutions for a model of non-sarcomeric actomyosin bundles
  • DCDS Home
  • This Issue
  • Next Article
    Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain
September  2016, 36(9): 4925-4943. doi: 10.3934/dcds.2016013

Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach

Weigao Ge 1, and  Li Zhang 2,

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2. 

Department of Foundation Courses, Beijing Union University, Beijing 100101, China

Received  August 2015 Revised  February 2016 Published  May 2016

By variational methods, this paper considers 4k-periodic solutions of a kind of differential delay systems with $2k-1$ lags. Our results reveal the fact that the number of $4k-$periodic orbits depends only upon the eigenvalues of both matrices $A_{\infty}$ and $A_0$. The conditions are more definite and easier to be examined. Moreover, two examples are given to illustrate the applications of the results.
Keywords: Periodic solution, critical point., variational approach, differential delay system.
Mathematics Subject Classification: Primary: 34K13; Secondary: 58E5.
Citation: Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013
References:
[1]

L. Fannio, Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity,, Discrete and Cont. Dynamicals Sys., 3 (1997), 251. doi: 10.3934/dcds.1997.3.251. Google Scholar

[2]

G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(I),, Nonlinear Anal., 65 (2006), 25. doi: 10.1016/j.na.2005.06.011. Google Scholar

[3]

G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(II),, Nonlinear Anal., 65 (2006), 40. doi: 10.1016/j.na.2005.06.012. Google Scholar

[4]

W. Ge, On the existence of periodic solutions of the differential delay equations with multiple lags,, Acta Appl. Math. Sinica(in Chinese), 17 (1994), 173. Google Scholar

[5]

W. Ge, Oscillatory periodic solutions of differential delay equations with multiple lags,, Chinese Sci. Bull., 42 (1997), 444. doi: 10.1007/BF02882587. Google Scholar

[6]

W. Ge, Periodic solutions of the differential delay equation $x'(t) = -f(x(t-1))$,, Acta Math. Sinica(New Series), 12 (1996), 113. doi: 10.1007/BF02108151. Google Scholar

[7]

W. Ge, Two existence theorems of periodic solutions for differential delay equations,, Chinese Ann. Math., 15 (1994), 217. Google Scholar

[8]

Z. Guo and J. Yu, Multiple results for periodic solutions to delay differential equations via critical point theory,, J. Differential Equations, 218 (2005), 15. doi: 10.1016/j.jde.2005.08.007. Google Scholar

[9]

Z. Guo and J. Yu, Multiple results on periodic solutions to higher dimensional differential equations with multiple delays,, J. Dynam. Differential Equations, 23 (2011), 1029. doi: 10.1007/s10884-011-9228-z. Google Scholar

[10]

J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solution of delay equations,, J. Math. Anal. Appl., 48 (1974), 317. doi: 10.1016/0022-247X(74)90162-0. Google Scholar

[11]

J. Li and X. He, Multiple periodic solutions of differential delay equations created by asymptotically linear Hailtonian systems,, Nonlinear Anal., 31 (1998), 45. doi: 10.1016/S0362-546X(96)00058-2. Google Scholar

[12]

J. Li and X. He, Proof and gengeralization of Kaplan-Yorke' conjecture under the condition $f'(0)>0$ on periodic solution of differential delay equations,, Sci. China Ser. A, 42 (1999), 957. doi: 10.1007/BF02880387. Google Scholar

[13]

J. Li, X. He and Z. Liu, Hamiltonian symmetric group and multiple periodic solutions of differential delay equations,, Nonlinear Analysis, 35 (1999), 457. doi: 10.1016/S0362-546X(97)00623-8. Google Scholar

[14]

S. Li and J. Liu, Morse theory and asymptotically linear Hamiltonian systems,, J. Differential Equations, 78 (1989), 53. doi: 10.1016/0022-0396(89)90075-2. Google Scholar

[15]

J. Mawhen and M. Willem, Critical Point Theory and Hamiltonian System,, Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[16]

R. D. Nussbaum, Periodic solutions of special differential delay equations: An example in nonlinear functional analysis,, Proc. Loyal Soc. Edingburgh, 81 (1978), 131. doi: 10.1017/S0308210500010490. Google Scholar

show all references

References:
[1]

L. Fannio, Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity,, Discrete and Cont. Dynamicals Sys., 3 (1997), 251. doi: 10.3934/dcds.1997.3.251. Google Scholar

[2]

G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(I),, Nonlinear Anal., 65 (2006), 25. doi: 10.1016/j.na.2005.06.011. Google Scholar

[3]

G. Fei, Multiple periodic solutins of differential delay equations via Hamiltonian systems(II),, Nonlinear Anal., 65 (2006), 40. doi: 10.1016/j.na.2005.06.012. Google Scholar

[4]

W. Ge, On the existence of periodic solutions of the differential delay equations with multiple lags,, Acta Appl. Math. Sinica(in Chinese), 17 (1994), 173. Google Scholar

[5]

W. Ge, Oscillatory periodic solutions of differential delay equations with multiple lags,, Chinese Sci. Bull., 42 (1997), 444. doi: 10.1007/BF02882587. Google Scholar

[6]

W. Ge, Periodic solutions of the differential delay equation $x'(t) = -f(x(t-1))$,, Acta Math. Sinica(New Series), 12 (1996), 113. doi: 10.1007/BF02108151. Google Scholar

[7]

W. Ge, Two existence theorems of periodic solutions for differential delay equations,, Chinese Ann. Math., 15 (1994), 217. Google Scholar

[8]

Z. Guo and J. Yu, Multiple results for periodic solutions to delay differential equations via critical point theory,, J. Differential Equations, 218 (2005), 15. doi: 10.1016/j.jde.2005.08.007. Google Scholar

[9]

Z. Guo and J. Yu, Multiple results on periodic solutions to higher dimensional differential equations with multiple delays,, J. Dynam. Differential Equations, 23 (2011), 1029. doi: 10.1007/s10884-011-9228-z. Google Scholar

[10]

J. Kaplan and J. Yorke, Ordinary differential equations which yield periodic solution of delay equations,, J. Math. Anal. Appl., 48 (1974), 317. doi: 10.1016/0022-247X(74)90162-0. Google Scholar

[11]

J. Li and X. He, Multiple periodic solutions of differential delay equations created by asymptotically linear Hailtonian systems,, Nonlinear Anal., 31 (1998), 45. doi: 10.1016/S0362-546X(96)00058-2. Google Scholar

[12]

J. Li and X. He, Proof and gengeralization of Kaplan-Yorke' conjecture under the condition $f'(0)>0$ on periodic solution of differential delay equations,, Sci. China Ser. A, 42 (1999), 957. doi: 10.1007/BF02880387. Google Scholar

[13]

J. Li, X. He and Z. Liu, Hamiltonian symmetric group and multiple periodic solutions of differential delay equations,, Nonlinear Analysis, 35 (1999), 457. doi: 10.1016/S0362-546X(97)00623-8. Google Scholar

[14]

S. Li and J. Liu, Morse theory and asymptotically linear Hamiltonian systems,, J. Differential Equations, 78 (1989), 53. doi: 10.1016/0022-0396(89)90075-2. Google Scholar

[15]

J. Mawhen and M. Willem, Critical Point Theory and Hamiltonian System,, Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[16]

R. D. Nussbaum, Periodic solutions of special differential delay equations: An example in nonlinear functional analysis,, Proc. Loyal Soc. Edingburgh, 81 (1978), 131. doi: 10.1017/S0308210500010490. Google Scholar

[1]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
[2]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
[3]

Fioralba Cakoni, Houssem Haddar. A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Problems & Imaging, 2007, 1 (3) : 443-456. doi: 10.3934/ipi.2007.1.443
[4]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[5]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[6]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[7]

Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164
[8]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
[9]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301
[10]

Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809
[11]

Arnaud Münch. A variational approach to approximate controls for system with essential spectrum: Application to membranal arch. Evolution Equations & Control Theory, 2013, 2 (1) : 119-151. doi: 10.3934/eect.2013.2.119
[12]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[13]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[14]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[15]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105
[16]

Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529
[17]

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
[18]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841
[19]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[20]

Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences