Abstract
A necessary and sufficient condition is established for the equilibrium of the damped superlinear oscillator
to be globally asymptotically stable. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation
is divergent or convergent. Since this nonlinear differential equation cannot be solved in general, it can be said that the presented result is expressed by an implicit condition. Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, it is proved that a certain growth condition of a(t) guarantees the global asymptotic stability for the equilibrium of the damped superlinear oscillator.
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References
Artstein, Z., Infante, E.F.: On the asymptotic stability of oscillators with unbounded damping. Quart. Appl. Math. 34, 195–199 (1976/77)
Bacciotti A., Rosier L.: Liapunov Functions and Stability in Control Theory. Springer, Berlin (2005)
Ballieu R.J., Peiffer K.: Attractivity of the origin for the equation \({\ddot{x} + f(t,\dot{x},\ddot{x})|\dot{x}|^{\alpha}\dot{x}\,+\, g(x) = 0}\) . J. Math. Anal. Appl. 65, 321–332 (1978)
Bass D.W., Haddara M.R.: Nonlinear models of ship roll damping. Int. Shipbuild. Prog. 35, 5–24 (1988)
Bass D.W., Haddara M.R.: Roll and sway-roll damping for three small fishing vessels. Int. Shipbuild. Prog. 38, 51–71 (1991)
Brauer, F., Nohel, J.: The Qualitative Theory of Ordinary Differential Equations. W.A. Benjamin, New York (1969); (revised) Dover, New York (1989)
Cardo A., Francescutto A., Nabergoj R.: On damping models in free and forced rolling motion. Ocean Eng. 9, 171–179 (1982)
Cesari, L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Springer, Berlin (1959); (2nd edn.) Springer, Berlin (1963)
Coppel W.A.: Stability and Asymptotic Behavior of Differential Equations. Heath, Boston (1965)
Dalzell J.F.: A note on the form of ship roll damping. J. Ship Res. 22, 178–185 (1978)
Duc L.H., Ilchmann A., Siegmund S., Taraba P.: On stability of linear time-varying second-order differential equations. Quart. Appl. Math. 64, 137–151 (2006)
Haddara M.R., Bass D.W.: On the form of roll damping moment for small fishing vessels. Ocean Eng. 17, 525–539 (1990)
Halanay A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York (1966)
Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969); (revised) Krieger, Malabar (1980)
Hatvani L.: On the asymptotic stability for a two-dimensional linear nonautonomous differential system. Nonlinear Anal. 25, 991–1002 (1995)
Hatvani L.: Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. Proc. Am. Math. Soc. 124, 415–422 (1996)
Hatvani L., Krisztin T., Totik V.: A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. Differ. Equ. 119, 209–223 (1995)
Hatvani L., Totik V.: Asymptotic stability of the equilibrium of the damped oscillator. Diff. Integr. Equ. 6, 835–848 (1993)
Hinemo, Y.: Prediction of Ship Roll Damping-State of Art (report no. 239). Department of Naval Architecture and Marine Engineering, The University of Michigan, Ann Arbor (1981)
Ignatyev A.O.: Stability of a linear oscillator with variable parameters. Electron. J. Differ. Equ. 1997(17), 1–6 (1997)
Karsai J., Graef J.R.: Attractivity properties of oscillator equations with superlinear damping. Discrete Contin. Dyn. Syst. 2005(suppl.), 497–504 (2005)
Levin J.J., Nohel J.A.: Global asymptotic stability for nonlinear systems of differential equations and application to reactor dynamics. Arch. Ration. Mech. Anal. 5, 194–211 (1960)
Michel A.N., Hou L., Liu D.: Stability dynamical systems: continuous, discontinuous, and discrete systems. Birkhäuser, Boston (2008)
Neves M.A.S., Pérez N.A., Valerio L.: Stability of small fishing vessels in longitudinal waves. Ocean Eng. 26, 1389–1419 (1999)
Pucci P., Serrin J.: Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math. 170, 275–307 (1993)
Pucci P., Serrin J.: Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. Anal. 25, 815–835 (1994)
Rouche N., Habets P., Laloy M.: Stability theory by Liapunov’s direct method. In: Applied Mathematical Sciences, vol. 22. Springer, New York (1977)
Smith R.A.: Asymptotic stability of \({x^{\prime\prime} + a(t)x^{\prime} + x = 0}\) . Quart. J. Math. 12, 123–126 (1961)
Sugie J.: Global asymptotic stability for damped half-linear oscillators. Nonlinear Anal. 74, 7151–7167 (2011)
Taylan M.: The effect of nonlinear damping and restoring in ship rolling. Ocean Eng. 27, 921–932 (2000)
Yoshizawa T.: Stability theory and the existence of periodic solutions and almost periodic solutions. In: Applied Mathematical Sciences, vol. 14. Springer, New York (1975)
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Sugie, J., Shimadu, T. & Yamasaki, T. Global Asymptotic Stability for Oscillators with Superlinear Damping. J Dyn Diff Equat 24, 777–802 (2012). https://doi.org/10.1007/s10884-012-9256-3
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DOI: https://doi.org/10.1007/s10884-012-9256-3
Keywords
- Damped oscillator
- Superlinear differential equations
- Global asymptotic stability
- Newtonian damping
- Growth condition