Asymptotics and stability of the delayed Duffing equation (Q1743804)

This paper studies the dynamics of the Duffing equations with a delay \[ x''(t)+\alpha_1 x'(t)+\alpha_2 x(t-\tau)+ \beta_1 x(t) + \beta_2 x(t-\tau)+ \gamma_1 x(t)^3 + \gamma_2x(t-\tau)^3=0. \] When the delay \(\tau=0\), the equation reduces to a standard Duffing equation whose dynamics is well-understood. The aim of the paper is to understand the behavior when the delay \(\tau>0\) is small. This requires the investigation of different cases according to the nature of the un-delayed system; in some cases the dynamics in the delayed case is equivalent to that of the undelayed case (structural stability), whereas in other cases new phenomena arise as result of the delay.