A novel route to a Hopf-bifurcation scenario in switched systems with dead zone

Planar switched system with dead-zone are analyzed. In particular, we consider the effects of perturbation of the linear control law from purely positional to position-velocity control. This type of perturbation leads to a novel Hopf-like discontinuity induced bifurcation. We show that this bifurcation leads to the creation of small scale limit cycle attractors, which scale as the square root of the bifurcation parameter. We note that a Hopf-like bifurcation analyzed in non-smooth systems is characterized by a linear scaling law. We then investigate numerically a planar switched system with positional feedback law, dead-zone and time delay in the switching function. Using the same parameter values as for the switched system without delay in the switching function, we show numerically a Hopf-like bifurcation scenario which matches qualitatively and quantitatively with the scenario analyzed for the non-delayed system.