Boundedness and stability for the damped and forced single well Duffing equation

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January 2013, 33(1): 211-223. doi: 10.3934/dcds.2013.33.211

Boundedness and stability for the damped and forced single well Duffing equation

Cyrine Fitouri ^1, and Alain Haraux ^1,

1.	UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France

Received August 2011 Revised January 2012 Published September 2012

Abstract
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By using differential inequalities we improve some estimates of W.S. LOUD for the ultimate bound and asymptotic stability of the solutions to the Duffing equation $ u''+ c{u'} + g(u)= f(t)$ where $c>0$, $f $ is measurable and essentially bounded, and $g$ is continuously differentiable with $g'\ge b>0$.

Keywords: Duffing equation., stability, Boundedness.

Mathematics Subject Classification: Primary: 34C11, 34C15,34C25, 34C27, 34D05; Secondary: 34D23, 34D3.

Citation: 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

References:

[1]	L. Amerio, Soluzioni quasi periodiche, o limitate, di sistemi differenziali non lineari quasi periodici, o limitati, Ann. Mat. Pura. Appl., 39 (1955), 97-119.
[2]	M. Biroli, Sur les solutions bornés et presque péiodiques des éuations et inéuations d'éolution, Ann. Mat. Pura Appl., 93 (1972), 1-79.
[3]	M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order, Ann. Math., 48 (1947), 472-494.
[4]	C. M. Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, in "Proccedings of a University of Florida International Symposium," Academic Press, 1977, 43-57.
[5]	C. Fitouri, "Thesis Dissertation," ch.2, University of Zürich, 2008.
[6]	C. Fitouri and A. Haraux, Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl., 92 (2009), 313-321.
[7]	A. Haraux, "Nonlinear Evolution Equations: Global Behavior of Solutions," Springer-Verlag, New York, 1981.
[8]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson,Paris, 1991.
[9]	A. Haraux, On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.,29 (2005), 207-230.
[10]	A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations, J. Dynam. Differential Equations, 19 (2007), 915-933.
[11]	W. S. Loud, On periodic solutions of Duffing's equation with damping, Journal of Mathematics and Physics,34 (1955), 173-178.
[12]	W. S. Loud, Boundedness and convergence of solutions of x''+cx' +g(x) = e(t), Duke Math. J., 24 (1957), 63-72.
[13]	W. S. Loud, Periodic solutions of x''+cx' +g(x) = f(t), Mem. Amer. Math. Soc., 31 (1959), 1-57.
[14]	Ph. Souplet, Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations, Nonlinear Analysis T.M.A., 26 (1996), 1511-1525. doi: 10.1016/0362-546X(95)00012-K.

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References:

[1]	L. Amerio, Soluzioni quasi periodiche, o limitate, di sistemi differenziali non lineari quasi periodici, o limitati, Ann. Mat. Pura. Appl., 39 (1955), 97-119.
[2]	M. Biroli, Sur les solutions bornés et presque péiodiques des éuations et inéuations d'éolution, Ann. Mat. Pura Appl., 93 (1972), 1-79.
[3]	M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order, Ann. Math., 48 (1947), 472-494.
[4]	C. M. Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, in "Proccedings of a University of Florida International Symposium," Academic Press, 1977, 43-57.
[5]	C. Fitouri, "Thesis Dissertation," ch.2, University of Zürich, 2008.
[6]	C. Fitouri and A. Haraux, Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl., 92 (2009), 313-321.
[7]	A. Haraux, "Nonlinear Evolution Equations: Global Behavior of Solutions," Springer-Verlag, New York, 1981.
[8]	A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson,Paris, 1991.
[9]	A. Haraux, On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.,29 (2005), 207-230.
[10]	A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations, J. Dynam. Differential Equations, 19 (2007), 915-933.
[11]	W. S. Loud, On periodic solutions of Duffing's equation with damping, Journal of Mathematics and Physics,34 (1955), 173-178.
[12]	W. S. Loud, Boundedness and convergence of solutions of x''+cx' +g(x) = e(t), Duke Math. J., 24 (1957), 63-72.
[13]	W. S. Loud, Periodic solutions of x''+cx' +g(x) = f(t), Mem. Amer. Math. Soc., 31 (1959), 1-57.
[14]	Ph. Souplet, Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations, Nonlinear Analysis T.M.A., 26 (1996), 1511-1525. doi: 10.1016/0362-546X(95)00012-K.

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