Periodic solutions for a class of Duffing differential equations

Amina Feddaoui; Jaume Llibre; Amar Makhlouf

doi:10.2478/mjpaa-2019-0007

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Periodic solutions for a class of Duffing differential equations

Moroccan Journal of Pure and Applied Analysis

Volume 5 (2019): Issue 1 (June 2019)

By: Amina Feddaoui, Jaume Llibre and Amar Makhlouf

Open Access

|Nov 2019

Abstract

We provide sufficient conditions for the existence of periodic solutions for the class of Duffing differential equations

$x^{″} + c (t) x^{'} + a (t) x + b (t) x^{3} = h (t, x),$ x'' + c(t)x' + a(t)x + b(t){x^3} = h(t,x),

where the functions a(t), b(t), c(t) and h(t, x) are C² and T–periodic in the variable t.

References

[1] R. Benterki and J. Llibre, Periodic solutions of the Duffing differential equation revisited via the averaging theory, J. of Nonlinear Modeling and Analysis 1 (2019), 10–26.
Search in Google Scholar Back to article
[2] R. Benterki and J. Llibre, Periodic solutions of a class of Duffing differential equations, to appear in J. of Nonlinear Modeling and Analysis.
Search in Google Scholar Back to article
[3] A. Buica, J.P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. Pure and Appl. Anal. 6 (2007), 103–111.10.3934/cpaa.2007.6.103
Search in Google Scholar Back to article
[4] 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), 3925–3932.10.1090/S0002-9939-07-09024-7
Search in Google Scholar Back to article
[5] H. Chen and Y. Li, Rate of decay of stable periodic solutions of Duffing equations, J. Differential Equations 236 (2007), 493–503.10.1016/j.jde.2007.01.023
Search in Google Scholar Back to article
[6] H. Chen and Y. Li, Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity 21 (2008), 2485–2503.10.1088/0951-7715/21/11/001
Search in Google Scholar Back to article
[7] A. C. Lazer and P. J. Mckenna, On the existence of stable periodic solutions of differential equations of Duffing type, Proc. Amer. Math. Soc. 110 (1990), 125–133.10.1090/S0002-9939-1990-1013974-9
Search in Google Scholar Back to article
[8] S. Liang, Exact multiplicity and stability of periodic solutions for a Duffing equation, Mediterr. J. Math. 10 (2013), 189–199.10.1007/s00009-012-0189-1
Search in Google Scholar Back to article
[9] J. Llibre and A. Makhlouf, Periodic solutions for periodic second-order differential equations with variable potentials, J Appl. Anal. 24 (2018), 127–137,10.1515/jaa-2018-0013
Search in Google Scholar Back to article
[10] J. Llibre and L. Roberto, On the periodic solutions of a class of Duffing differential equations, Discrete Contin. Dyn. Syst. 33 (2013), 277–282.10.3934/dcds.2013.33.277
Search in Google Scholar Back to article
[11] I.G. Malkin, Some Problems of the theory of nonlinear oscillations, Gosudarstv. Izdat. Tehn-Teor. Lit. Moscow, 1956 (in Russian).
Search in Google Scholar Back to article
[12] J. Mawhin, Seventy-five years of global analysis around the forcedpendulum equation, in Equadiff, Brno. Proceedings, Brno: Masaryk University 9 (1997), 115–145.
Search in Google Scholar Back to article
[13] F.I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, Appl. Math. Comput 135 (2003), 471–490.10.1016/S0096-3003(02)00062-0
Search in Google Scholar Back to article
[14] R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boo. Uni. Mat. Ital B 3 (1989), 533–546.
Search in Google Scholar Back to article
[15] M. Roseau, Vibrations non linéaires et théorie de la stabilité, (French) Springer Tracts in Natural Philosophy, Vol. 8 Springer-Verlag, Berlin-New York, 1966.
Search in Google Scholar Back to article
[16] J.A. Sanders, F. Verhulst and J. Murdock, Averaging method in nonlinear dynamical systems, Appl. Math. Sci., Vol. 59, Springer, New York, 2007.
Search in Google Scholar Back to article
[17] P.J. Torres, Existence and stability of periodic solutions of a Duffing equation by using a new maximum principale, Mediterr. J. Math. 1 (2004), 479–486.10.1007/s00009-004-0025-3
Search in Google Scholar Back to article
[18] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Universitext, Springer, New York, 1996.10.1007/978-3-642-61453-8
Search in Google Scholar Back to article
[19] F. Wang and H. Zhu, Existence, uniqueness and stability of periodic solutions of a Duffing equation under periodic and anti-periodic eigenvalues conditions, Taiwanese J of Math. 19 (2015), 1457–1468.10.11650/tjm.19.2015.3992
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/mjpaa-2019-0007 | Journal eISSN: 2351-8227

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Language: English

Page range: 86 - 103

Submitted on: Aug 31, 2019

Accepted on: Oct 6, 2019

Published on: Nov 8, 2019

Published by: Sciendo

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Periodic solution,

averaging method,

Duffing differential equation

Related subjects:

Mathematics,

Analysis,

Differential equations and dynamical systems,

Probability and statistics,

Numerical and computational mathematics

© 2019 Amina Feddaoui, Jaume Llibre, Amar Makhlouf, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 5 (2019): Issue 1 (June 2019)