American Institute of Mathematical Sciences

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

 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

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 $g'\ge b>0$.
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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