# Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA)

**CICADA GRADUATE LECTURES**

**An Introduction to Modelling using Hybrid Systems**

**Total Time 20 hours over 10 weeks + 10 tutorial sessions**

Venue: Frank Adams 2, Alan Turing Building

The lectures on "Simulation of Biological Systems have been moved forward to begin on Monday 19th March with the first two lectures. The third lecture will be given on Thursday 22nd March in place of the tutorial session.

The lectures will then recommence AFTER the Easter break with the lectures on Model Simplification beginning on Monday April 16th.

Lecture Notes and Additional Materials

**Purpose of the course**

The course aims to supply students interested in applying mathematical modelling to systems that display “hybrid” characteristics, for example systems that have both continuous and discrete components. A very simple example of such a system is a thermostat. The course will present examples of systems from engineering and from the biosciences. The aim is to cover sufficicnt mathematical background to enable the students to pursue further research in hybrid systems. The course will be examples based, it will concentrate primarily on the tools for modelling hybrid systems.

**Potential audience**

Students studying both pure and applied mathematics and who are interested in approaches which combine elements of both. Students studying engineering or biological science will find the mathematical techniques and examples highly relevant to their discipline. Hybrid systems are involved in control of both mechanical and biological systems. Insights from both applied and pure mathematics can provide tools to understand such systems in new ways. Specialist mathematical skills are not essential, students will be able to pursue the topics introduced in a variety of ways, some closer to the physical understanding of the topic, some more mathematical and formal in nature.

**Structure of the course**

There will be 20 lectures and 10 tutorials. The lectures will be on Mondays from 12:00 to 14:00 and the tutorials will be on Thursdays at 12:00 to 13:00. Lectures will start on Monday 13th February and will continue until March 19th. They will restart on Monday 16th April and continue until the week ending 11th May (Monday lecture will be moved to Tuesday for this week owing to Bank Holiday.)

**CICADA and course tutors**

The course tutors are all involved in the CICADA project. Details of CICADA and the research interests of the tutors can be found at

http://www.cicada.manchester.ac.uk

**Registration**

Registration of students will take place during the first class of the lecture series on the 13th February, so please just turn up on the day.

**Reading list **

Each topic will have a reading list supplied by the tutors for each topic and introduced in the lectures.

**Syllabus**

**1. Examples and Intoduction**

John Brooke/Paul Glendinning + all tutors 2 hours

Examples of hybrid systems, the types of problems and questions and the difficulty of using ‘standard’ tools. Overview of the topics to be addressed in the course.

**2. Hybrid Systems**

Paul Glendinning/John Brooke - 2 hours

Different spaces (discrete/continuous etc), ideas of time, more abstract definition of hybrid systems – guards, control, synchronous/asynchronous time, stochasticity etc.

**3. Dynamics and Stability**

James Hook/Paul Glendinning - 3 hours

Simple examples of ways switches change behaviour, more examples, Indication of general approaches. Scalability/lack of scalability (what we can do rather than what we would like to do!).

**4. Verification**

Gareth Jones - 3 hours

Reachability/control, model checking, including bisimulation. Analytical, approximation and simulation techniques.

**5. Model Simplification**

John Brooke - 3 hours

The continuous part of a hybrid system is often modeled by a dynamical system infinite dimension. In order to simulate this numerically the problem must be discretized and if possible the dimension of the state space reduced. Topics covered: discretization for simulation, discretization methods (e.g. Markov), Large scale modelling: Model reduction and control of finite dimensional systems derived from discretization of infinite dimensional systems.

**7. Simulation of Biological Systems**

Jürgen Pahle - 3 hours

This theme will present biological systems as examples of hybrid systems and will demonstrate how such systems can be simulated.

**8. Algebraic Techniques**

Marianne Johnston, Mark Muldoon, Mark Kambites - 4 hours

Maxplus, Petri nets (discrete space and cts time), large reaction networks, connections with optimization and hybrid models.