Two-dimensional map for impact oscillator with drift

An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau, relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.