Chaotic and random point processes: analysis, design and application to hybrid systems

Processes characterised by point events occurring in time are called point processes. Emissions from a radioactive source, action potential in a nerve fibber, traffic flow passing through a designated location on the highway, are all examples of pint processes. In our tutorial we firstly discuss point processes associated with deterministic chaotic dynamics of hybrid systems such a5 power electronic circuits, systems involving switches, impacting mechanical systems, continuous systems controlled by discrete logic (aircraft auto-pilot modes, thermostats, chemical plants with on-off valves and pumps, automobile automatic transmissions). An understanding of bifurcations (qualitative change in a system's dynamical behaviour) is very important for the operation and control of any system. In this tutorial, we seek to develop a theoretical understanding of the bifurcation phenomena that occur in hybrid systems. We'll show that such systems yield piecewise smooth maps under sampled-data modelling. Next, we'll introduce various smooth and non-smooth bifurcations that can happen in such systems. In presenting the border collision bifurcations, we'll review the various approaches to the problem and present a synthetic theoretical framework that can be readily applied in explaining bifurcation phenomena in hybrid systems. Then we present a systematic approach to the statistical analysis, control and design of hybrid dynamical systems with chaotic behaviour in terms of the statistical properties of