Synchronization of solutions of Duffing-type equations with random perturbations

We consider a family of particles with different initial states and/or velocities whose dynamics is described by a Duffing-type equation ẍ+αẋ+P(x)=0 where the velocity v=ẋ is randomly perturbed at random times. We present sufficient conditions ensuring almost identical sample paths of the particles after a long time: the times between perturbations are assumed to be unbounded and uniformly positive, and the values of jumps are assumed to be random variables with positive density on a sufficiently large interval [0,H][0,H]. The paper generalizes the results of Ambrazevičius et al. (2010)  [11] for P(x)=x3−xP(x)=x3−x to the case of arbitrary higher-order odd polynomials PP.