# 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 & Continuous Dynamical Systems - A, 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.   Google Scholar [2] M. Biroli, Sur les solutions bornés et presque péiodiques des éuations et inéuations d'éolution,, Ann. Mat. Pura Appl., 93 (1972), 1.   Google Scholar [3] M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order,, Ann. Math., 48 (1947), 472.   Google Scholar [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.   Google Scholar [5] C. Fitouri, "Thesis Dissertation," ch.2,, University of Zürich, (2008).   Google Scholar [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.   Google Scholar [7] A. Haraux, "Nonlinear Evolution Equations: Global Behavior of Solutions,", Springer-Verlag, (1981).   Google Scholar [8] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (1991).   Google Scholar [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.   Google Scholar [10] A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations,, J. Dynam. Differential Equations, 19 (2007), 915.   Google Scholar [11] W. S. Loud, On periodic solutions of Duffing's equation with damping,, Journal of Mathematics and Physics, 34 (1955), 173.   Google Scholar [12] W. S. Loud, Boundedness and convergence of solutions of x''+cx' +g(x) = e(t),, Duke Math. J., 24 (1957), 63.   Google Scholar [13] W. S. Loud, Periodic solutions of x''+cx' +g(x) = f(t),, Mem. Amer. Math. Soc., 31 (1959), 1.   Google Scholar [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.  doi: 10.1016/0362-546X(95)00012-K.  Google Scholar

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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.   Google Scholar [2] M. Biroli, Sur les solutions bornés et presque péiodiques des éuations et inéuations d'éolution,, Ann. Mat. Pura Appl., 93 (1972), 1.   Google Scholar [3] M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order,, Ann. Math., 48 (1947), 472.   Google Scholar [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.   Google Scholar [5] C. Fitouri, "Thesis Dissertation," ch.2,, University of Zürich, (2008).   Google Scholar [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.   Google Scholar [7] A. Haraux, "Nonlinear Evolution Equations: Global Behavior of Solutions,", Springer-Verlag, (1981).   Google Scholar [8] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,", Masson, (1991).   Google Scholar [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.   Google Scholar [10] A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations,, J. Dynam. Differential Equations, 19 (2007), 915.   Google Scholar [11] W. S. Loud, On periodic solutions of Duffing's equation with damping,, Journal of Mathematics and Physics, 34 (1955), 173.   Google Scholar [12] W. S. Loud, Boundedness and convergence of solutions of x''+cx' +g(x) = e(t),, Duke Math. J., 24 (1957), 63.   Google Scholar [13] W. S. Loud, Periodic solutions of x''+cx' +g(x) = f(t),, Mem. Amer. Math. Soc., 31 (1959), 1.   Google Scholar [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.  doi: 10.1016/0362-546X(95)00012-K.  Google Scholar
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