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

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