Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms

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; ? _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 ? _p (x′), ? _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.