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

TL;DR

This paper investigates a new type of Hopf-like bifurcation in planar switched systems with dead zones, caused by perturbations from positional to position-velocity control, leading to small limit cycles.

Contribution

It introduces a novel Hopf-like bifurcation scenario in non-smooth systems due to control law perturbations, with numerical validation including time delay effects.

Findings

Bifurcation leads to small limit cycles scaling with the square root of the bifurcation parameter.

The Hopf-like bifurcation exhibits linear scaling law characteristic of non-smooth systems.

Numerical simulations confirm the bifurcation scenario with and without time delay.

Abstract

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.