Criticality is one of the most important properties of Complex Systems. Criticality occurs in two related but distinct ways: 1. when a system unexpectedly collapses from one state to another, very different, state and 2. when a system is in a state with wild fluctuations and it is highly sensitive to small changes in behaviours. In the first case we call it a 'Tipping Point' and in the second case a 'Continuous Critical Transition'. There are many practical examples of these types of behaviour: Financial markets are often in a continuous critical transition and they can also quickly collapse, diseases that are transferred through a social network can suddenly explode into a pandemic, and a local power outage in an electricity network can cause entire cities to blackout. We will also look at selforganised criticality, where a system evolves to be near one of these 'dangerous' critical points, this is one of the most exciting emergent phenomena in modern applied sciences, engineering and business and we will cover present several real-world applications in this area. This unit will study a range of important examples in which criticality plays a key role and we will show what the underlying causes are for these uncontrolled collapses and wild dynamics. We will use a combination of software examples (Matlab) and mathematical techniques in order to illustrate when and how such interactions might occur and how to simulate their dynamics. It will cover crossdisciplinary concepts and methods based on nonlinear dynamics, including elements of chaos theory and statistical physics, such as fractals and percolation.
Refer to the assessment table in the unit outline.
Mathematics at first-year undergraduate level. Some familiarity with mathematical and computational principles at an undergraduate university level (for example, differential calculus or linear algebra). Familiarity with a programming language at a beginners level for data analysis.