Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms
Tóm tắt
This paper deals with the damped superlinear oscillator
$$x'' + a(t)\phi_p\bigl(x' \bigr) + b(t)\phi_q\bigl(x'\bigr)+ \omega^2x = 0, $$
where a(t) and b(t) are continuous and nonnegative for t≥0; p and q are real numbers greater than or equal to 2; ϕ
r
(x′)=|x′|
r−2
x′. This equation is a generalization of nonlinear ship rolling motion with Froude’s expression, which is very familiar in marine engineering, ocean engineering and so on. Our concern is to establish a necessary and sufficient condition for the equilibrium to be globally asymptotically stable. In particular, the effect of the damping coefficients a(t), b(t) and the nonlinear functions ϕ
p
(x′), ϕ
q
(x′) on the global asymptotic stability is discussed. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation
$$u' + \omega^{p-2}a(t)\phi_p(u) + \omega^{q-2}b(t)\phi_q(u) + 1 = 0 $$
is divergent or convergent. In addition, explicit sufficient conditions and explicit necessary conditions are given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, some examples are included to illustrate our results. Finally, our results are extended to be applied to a more complicated model.
Tài liệu tham khảo
Artstein, Z., Infante, E.F.: On the asymptotic stability of oscillators with unbounded damping. Q. Appl. Math. 34, 195–199 (1976/77)
Bacciotti, A., Rosier, L.: Lyapunov Functions and Stability 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)
Berg, M.: A three-dimensional airspring model with friction and orifice damping. Veh. Syst. Dyn. 33, 528–539 (1999)
Brauer, F., Nohel, J.: The Qualitative Theory of Ordinary Differential Equations. Benjamin, New York (1969). Dover, New York (1989) (revised)
Bulian, G.: Approximate analytical response curve for a parametrically excited highly nonlinear 1-DOF system with an application to ship roll motion prediction. Nonlinear Anal., Real World Appl. 5, 725–748 (2004)
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). Springer, Berlin (1963) (2nd ed.)
Chan, H.S.Y., Xu, Z., Huang, W.L.: Estimation of nonlinear damping coefficients from large-amplitude ship rolling motions. Appl. Ocean Res. 17, 217–224 (1995)
Chun, H.H., Chun, S.H., Kim, S.Y.: Roll damping characteristics of a small fishing vessel with a central wing. Ocean Eng. 28, 1601–1619 (2001)
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. Q. Appl. Math. 64, 137–151 (2006)
Grace, I.M., Ibrahim, R.A., Pilipchuk, V.N.: Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations. Nonlinear Dyn. 66, 589–607 (2011)
Haddara, M.R.: On the stability of ship motion in regular oblique waves. Int. Shipbuild. Prog. 18, 416–434 (1971)
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). Krieger, Malabar (1980) (revised)
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., Totik, V.: Asymptotic stability of the equilibrium of the damped oscillator. Differ. Integral Equ. 6, 835–848 (1993)
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)
Hinemo, Y.: Prediction of ship roll damping-state of art. Dept. of Naval Architecture and Marine Engineering, The University of Michigan, Report No. 239 (1981)
Ibrahim, R.A., Grace, I.M.: Modeling of ship roll dynamics and its coupling with heave and pitch. Math. Probl. Eng. 2010, 1–32 (2010). Article ID 934714
Ignatyev, A.O.: Stability of a linear oscillator with variable parameters. Electron. J. Differ. Equ. 1997(17), 1–6 (1997)
Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Dekker, New York (1989)
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 Lyapunov’s Direct Method. Applied Mathematical Sciences, vol. 22. Springer, New York (1977)
Senjanović, I., Parunov, J., Ciprić, G.: Safety analysis of ship rolling in rough sea. Chaos Solitons Fractals 8, 659–680 (1997)
Shimozawa, K., Tohtake, T.: An air spring model with non-linear damping for vertical motion. Q. Rep. RTRI 49, 209–214 (2008)
Smith, R.A.: Asymptotic stability of x″+a(t)x′+x=0. Q. J. Math. 12, 123–126 (1961)
Sugie, J.: Global asymptotic stability for damped half-linear oscillators. Nonlinear Anal. 74, 7151–7167 (2011)
Sugie, J.: Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping. Proc. Am. Math. Soc. 141, 2419–2427 (2013)
Sugie, J., Hata, S.: Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients. Monatshefte Math. 166, 255–280 (2012)
Sugie, J., Shimadu, T., Yamasaki, T.: Global asymptotic stability for oscillators with superlinear damping. J. Dyn. Differ. Equ. 24, 777–802 (2012)
Taylan, M.: The effect of nonlinear damping and restoring in ship rolling. Ocean Eng. 27, 921–932 (2000)
Üçer, E.: Examination of the stability of trawlers in beam seas by using safe basins. Ocean Eng. 38, 1908–1915 (2011)
Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Applied Mathematical Sciences, vol. 14. Springer, New York (1975)