The occurrence of discontinuous right hand sides in ODE-systems often appears in technical applications. Such applications may be characterised by the cases where the system changes between several states. Each state is defined by a system of ODEs and the transition between states is defined by an algebraic condition. The numerical solution that is done in order to simulate the behaviour of the system will be possible by using standard numerical software but this approach is very ineffecient. We present an alternative approach based upon the tracking of state-changes and accurate numerical determination of transition points. Real applications are used to illustrate the approach.