School of Mathematics and Statistics

Complex systems and chaos


Prof Michael Small
Telephone (+61 8) 6488 3877


This will be an introduction to various areas in modern applied mathematics.  Each lecture will cover a separate topic from the areas of chaos and complex systems.  Topics will be covered in the following order.

How long is a piece of string?

The answer:  it depends.  It depends on how long your ruler is and how wiggly your string.  The dimension of objects turns out to generalise well beyond our intuitive understanding of 1, 2 and 3 dimensional space - not just to higher dimensions, but also to fractional dimensions.  I will talk about how these measures can be defined and employed for a variety of problems from measuring the length of coastline, to quantifying abstract art, branching structures in bronchiole and trees (the green leafy type not the ones that graph theorists talk about), lung disease and fluctuations in the stock market.  We will start with Iterated Function Systems.   

Unleashing chaos

Chaotic dynamics has many, slightly varying mathematical definitions - loosely, we mean bounded, deterministic and non-periodic behaviour.  That is, the state of the systems is always bounded by some 'box', the systems obeys deterministic equations and the dynamics are not periodic or a fixed point.  Chaos `looks random' but it isn't - it can be precisely described mathematically.  A simple clock pendulum (ignore friction as mathematicians tend to do) is periodic.  However, give the pendulum a second pivot and it becomes chaotic. Low-dimensional chaotic dynamics arise in many different and important physical systems - dynamics in atmospheric weather forecast, individual neurones, dynamics of common diseases.  We will talk about some of these examples and how understanding chaotic dynamics can be useful.

Your friends have more friends than you

The title is a statement of 'The Friendship Paradox', which is of course, not a paradox but only an apparent paradox.  I will explain this paradox (hint: it is due to a form of sampling bias) and other interesting features of social networks.  The networks we consider are large irregular graphs and they can be formulated as models of all sorts of interaction from Facebook contacts and co-publication in academic journals to neuronal networks and international air transport.  We will look at where these networks come from, how they grow and a few of their more interesting properties. 

Linear ordinary differential equations can make you rich

Along the way to the development of modern chaos theory, several of the early protagonists in this story took an interest in applying their skill and knowledge to games of chance. Gambling games have been a useful source of homework questions for high school, and first year, probability.   However, many of these systems are actually, in part, deterministic and that determinism can be employed to gain an edge on the house.  I will provide details of how this can be done with the example of Roulette. 

Additional information about complex systems, chaos, fractal dimension, newscientist and network theory is available for you.

There will be four mini-classes 45 minutes in duration given at 1pm on Tuesdays and Fridays in the Blakers Lecture Theatre.  No registration is required, simply turn up on your preferred date listed below.



Complex systems and chaos : Prof Michael Small
Prof Michael Small

bout the lecturer:  Michael is an applied mathematician with interests in nonlinear dynamical systems theory, complex systems theory, chaos, nonlinear time series analysis, and complex networks.  His research has found application in a diverse range of areas: physiology, neuroscience, ecology, granular systems, social networks, disease transmission, finance, gambling, musical composition and engineering for remote operations. 


Week Dates Lecturer Topic
8 15 September Prof Michael Small Complex systems and chaos


Week Dates Lecturer Topic
9, 10, 11 25 September, 9, 16 October Prof Michael Small Complex systems and chaos

Lecture notes


School of Mathematics and Statistics

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Last updated:
Sunday, 18 October, 2015 8:47 AM