Construction of asymptotic solutions for quasilinear third-order differential equations with lag (Q790334)

The paper deals with small perturbations of the linear differential equation with time lag \[ (1)\quad x'''(t)+\alpha_ 1x''(t)+\alpha_ 2x'(t)+\alpha_ 3x(t)+\beta_ 1x''(t-\Delta)+\beta_ 2x'(t- \Delta)+\beta_ 3x(t-\Delta)=0, \] where \(\Delta>0\), \(\alpha_ j\) and \(\beta_ j\) are constants. Under the assumption that the corresponding characteristic equation possesses one pair of purely imaginary roots \(\pm i\omega\), while the remaining roots have negative real parts with sufficiently large absolute values, the author looks for solutions of the quasilinear equation corresponding to (1) which are close to the solutions of the equation (1) of the form a co\(s(\omega\) t\(+\psi) (a,\psi\in R)\). The autonomous case, the nonautonomous case without resonance and the nonautonomous case with resonance are treated.