Asymptotic behavior and its visualization of the solutions of intermittently and impulsively damped nonlinear oscillator equations
This paper is concerned with the asymptotic behavior of solutions in some second-order oscillator equations. Two types of solutions are considered: The first type is the so-called intermittently damped equation given by \(x''+ g(t)x'+ f(x)= 0\) with \(f(x)= | x|^\alpha\text{sgn}(x)\), \(\alpha\) is a positive constant and \(g(t)\) a step function defined by three sequences \((t_n)\), \((i_n)\), \((g_n)\) so that \(g(t)= g_n\), for \(t\in [t_n, t_n+ i_n]\) and \(g(t)= 0\) otherwise. The second type is the impulsively damped equation \(x''+ f(x)= 0\) in which all solutions \(x(t)\) are damped at a sequence of points \((t_n)\) in the velocity so that \(x(t_n+ 0)= x(t_n)\), \(x'(t_n+ 0)= b_nx'(t_n)\). For each of these problems and for \(\alpha= 1\) (linear), \(\alpha>1\) (superlinear) and \(\alpha<1\) (sublinear), some conditions on the parameters that define the damping are given. They imply the stability and asymptotic stability of the solutions. This study is based mainly on the behavior along the considered solution of the energy function associated to the differential equation.
scientific article; zbMATH DE number 1183194