• 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

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.
Citation: Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete and Continuous Dynamical Systems, 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), 3925-3932.

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 21 (2008), 2485-2503.

[3]

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

[4]

J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation, in: "Equadiff, Brno. Proceedings, Brno: Masaryk University,'' 9 (1997), 115-145.

[5]

R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type, Boo. Uni. Mat. Ital B, 3 (1989), 533-546.

[6]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'' Universitext, Springer, 1991.

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), 3925-3932.

[2]

H. B. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 21 (2008), 2485-2503.

[3]

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

[4]

J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation, in: "Equadiff, Brno. Proceedings, Brno: Masaryk University,'' 9 (1997), 115-145.

[5]

R. Ortega, Stability and index of periodic solutions of an equation ofDuffing type, Boo. Uni. Mat. Ital B, 3 (1989), 533-546.

[6]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,'' Universitext, Springer, 1991.

[1]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961

[10]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]