A Negative Bendixson-Like Criterion for a Class of Hybrid Systems

A condition which ensures the absence of periodic orbits for nonsmooth dynamical systems is presented. The condition is a higher dimensional generalization of Bendixson's criterion applicable to differential inclusions that are useful in the description of hybrid systems. The main argument is based on contraction analysis of the d-measured volume along the system trajectories. A connection to methods for estimating the Hausdorff dimension is emphasized. For a class of hybrid systems described by a linear system with relay feedback the conditions are presented in the form of linear matrix inequalities. A simple but illustrative example is analyzed