Controlling the dangerous effect of the stochastically vibrating border in systems modeled by piecewise smooth 1-D maps

Abstract

Random vibration of the border in one dimensional piecewise smooth maps may lead to the occurrence of non-deterministic dynamics. Further, this type of non-deterministic dynamics may result an unexpected sudden collapse of such systems, if the variation of the border remains unnoticed. Therefore it is important to suggest some control method to avoid such possible danger. In this paper, we have suggested an effective way of controlling the existence of the non-deterministic dynamics occurring due to the random vibration of the border in one-dimensional piecewise smooth maps. The proposed control is achieved via a periodic time-dependent switching in a specific compartment of the phase space. In presence of this kind of switching, the resulting systems become time-dependent one-dimensional piecewise smooth maps. We have shown that this type of systems always posses a stable fixed point attractor and the dynamics of any orbit becomes deterministic. Therefore the proposed control method ensures an effective technique to overcome the possible danger. Finally, we have also derived a sufficient condition of convergence of an orbit to the existing stable fixed point attractor of the controlled system. This condition shows that we can pre-assign the period of the time-dependent switching judiciously in order to control the system dynamics.