The equivalence between feedback and dissipation in impact oscillators

We analyze an important class of engineering systems characterized by the discontinuous motion of a spring-mass constrained by the motion of a feedback-assisted actuator. We show that the combined effects of mechanical restitution coefficient and displacement feedback can be exactly represented by a single equivalent dissipation coefficient. We also show that the topological properties of the surfaces of section of orbits generated by impact oscillators which possess differing proportions of restitution and feedback levels, but whose equivalent dissipation coefficients are equal, are equivalent and universally scalable. The scaling law allows us to interchange the effects of restitution and feedback coefficients and so, effectively, eliminate one of these parameters from the equations of motion. Thus, the topological properties of dissipative feedback-assisted systems can be seen as scaled versions of either purely dissipative, or purely feedback-assisted, oscillators.