American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Generalized linear differential equations in a Banach space: Continuous dependence on a parameter
  • DCDS Home
  • This Issue
  • Next Article
    Existence and enclosure of solutions to noncoercive systems of inequalities with multivalued mappings and non-power growths
January  2013, 33(1): 277-282. doi: 10.3934/dcds.2013.33.277

On the periodic solutions of a class of Duffing differential equations

Jaume Llibre 1, and  Luci Any Roberto 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2. 

Departamento de Matemática, Ibilce -UNESP, 15054-000 São José do Rio Preto, Brazil

Received  February 2011 Revised  December 2011 Published  September 2012

In this work we study the periodic solutions, their stability and bifurcation for the class of Duffing differential equation $x''+ \epsilon C x'+ \epsilon^2 A(t) x +b(t) x^3 = \epsilon^3 \Lambda h(t)$, where $C>0$, $\epsilon>0$ and $\Lambda$ are real parameter, $A(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions and $\epsilon$ is sufficiently small. Our results are proved using the averaging method of first order.
Keywords: bifurcation, stability., Duffing differential equation, averaging method, Periodic solution.
Mathematics Subject Classification: Primary: 34C23, 34C25, 34C29, 34D20, 34G1.
Citation: Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
References:
[1]

H. B. Chen and Y. Li, Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities,, Proc. Amer. Math. Soc., 135 (2007), 3925.   Google Scholar

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations,, Nonlinearity, 21 (2008), 2485.   Google Scholar

[3]

G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung,, Sammlung Viewg Heft, 41/42 (1918).   Google Scholar

[4]

J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation,, in:, 9 (1997), 115.   Google Scholar

[5]

R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type,, Boo. Uni. Mat. Ital B, 3 (1989), 533.   Google Scholar

[6]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'', Universitext, (1991).   Google Scholar

show all references

References:
[1]

H. B. Chen and Y. Li, Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities,, Proc. Amer. Math. Soc., 135 (2007), 3925.   Google Scholar

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations,, Nonlinearity, 21 (2008), 2485.   Google Scholar

[3]

G. Duffing, Erzwungen Schwingungen bei vernäderlicher Eigenfrequenz undihre technisch Bedeutung,, Sammlung Viewg Heft, 41/42 (1918).   Google Scholar

[4]

J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation,, in:, 9 (1997), 115.   Google Scholar

[5]

R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type,, Boo. Uni. Mat. Ital B, 3 (1989), 533.   Google Scholar

[6]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'', Universitext, (1991).   Google Scholar

[1]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[2]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
[3]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[4]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[5]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[6]

Cyrine Fitouri, Alain Haraux. Boundedness and stability for the damped and forced single well Duffing equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 211-223. doi: 10.3934/dcds.2013.33.211
[7]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
[8]

Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779
[9]

Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
[10]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[11]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[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]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[14]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[15]

Zhaosheng Feng, Goong Chen, Sze-Bi Hsu. A qualitative study of the damped duffing equation and applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1097-1112. doi: 10.3934/dcdsb.2006.6.1097
[16]

S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660
[17]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[18]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[19]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[20]

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

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences