Controlling chaos of a dynamical system with discontinuous vector field

The paper presents how to control the chaos of dynamical systems with discontinuous vector field through the paradigm of a harmonically forced oscillator having a set-up elastic stop. It is first shown that the Poincaré mapping of this oscillator is not smooth near the fixed point corresponding to a periodic motion that grazes the stop. Thus, the current control strategies based on the smooth mapping cannot be directly used to stabilize the chaotic motion near a periodic grazing motion. Then proposed is a piecewise-linear control strategy based on the piecewise-linearized Poincaré mapping reconstructed from sampled data and on the pole assignment in two regions near the fixed point. The efficacy of the strategy is finally demonstrated via numerical simulations.