Bifurcation and Chaotic Behaviors in a Discrete Variable Structure System with Unbounded Control Magnitude

Bifurcation and chaotic behaviors in a discrete variable structure system with unbounded control magnitude are examined in this paper. The analysis using the exponential discretization shows that the sampling period is actually an additional degree of freedom whose value is crucial to the occurrence of bifurcation and chaotic behaviors. The bifurcation point of the sampling period is derived. Moreover, it is shown that other commonly used approximate discretization schemes such as zero-order-hold may cause similar problems as well. The bifurcation points of sampling periods for these schemes are an approximation of that for the exponential discretization scheme. Computer simulations are presented to illustrate the bifurcation and chaotic phenomena.