# Nonlinear and Complex Systems Research Group

## Research programmes

### Aspects of Hamiltonian Dynamics

The symmetric barrier billiard system was first introduced by Wiersig, and is an example of a pseudo-integrable dynamical system: a system which consists of multiple copies of an integrable Hamiltonian system, but which itself is not integrable. The motion consists of a particle moving at constant speed inside a rectangle experiencing specular collisions with the boundary (think of a screensaver), but with a partial barrier placed centrally which splits the configuration space into two chambers. It can be shown that the system can be completely understood by studying the transitions between these two chambers, and this leads to analysis of the autocorrelation function. For a particle moving at with slope equal to the golden mean, we have shown using a renormalization analysis that the correlations at Fibonacci times chaotically explore an invariant surface. Our current aim is to construct a model space which completely describes the behaviour of the renormalization operator which gives rise to this phenomena.

### Chaos and integrability from an algebraic viewpoint

In the realm of Dynamical Systems, an important problem is to decide whether a given system is “solvable” in a reasonably simple manner. If so, the system is usually called “integrable”. Otherwise, its evolution with respect to time is generally unpredictable and very sensitive to initial conditions (a phenomenon commonly known as chaos). One concept is not intrinsically antonymous with the other, but there seems to be indeed an inverse correlation between the two in practical examples. We currently use Galois theory, monodromy, Stokes’ theorem, and more recently formal calculus and infinite dimensional linear operators as tools to find intrinsic algebraic structures and settings characterising dynamical systems.

### Gilbert tessellations

A Gilbert tessellation is created when cracks begin to form at a set of points randomly spread throughout the plane. It is a simple model for the growth of needle shaped crystals, but its properties have remained resistant to mathematical analysis for decades. Recent work, in collaboration with Professor Richard Cowan (University of Sydney), has focussed on a Gilbert Tessellation which we call the "half rectangular" model which has surprisingly similar properties to the full rectangular model. The "half" model differs from the "full" in that East growing cracks cannot see South, and West cannot see North. Amongst other things, we have found that the average length of a ray in this model is π /Γ(3/4)^{2}.

### Geometry and chaos

Geodesics are the “straight lines” of curved surfaces, and even on apparently simple surfaces the behaviour of geodesics can be very complex. Recent work established the chaotic nature of geodesics on surfaces defined in terms of spherical harmonics, and current work focuses on the conjugate locus and in particular its bifurcations.

### Mathematical biology

Mathematical models are of major importance in understanding biological and medical systems. Infectious diseases, plant-soil-atmosphere interactions, cancer cells dynamics are all examples of successful synergy between mathematics and biological understanding. Analysis of these models reveals the connections between system components, the effects of these connections, and predicts how the system will evolve if changes in its components occur.

### Network structure and collective behaviour

From ecosystems to chemical reactors to computer networks, many natural and artificial systems have a network structure - individual elements communicate with each other and some collective behaviour emerges. But to what extent does the network architecture of such a system determine whether behaviours such as "switching", oscillation, or chaos are allowed? In fact, using analytical, algebraic and geometric approaches, a surprising amount can be said about the dynamics of networks from the network structure alone, and without computer simulation.

### Physics of super cooled gases

On the smallest length scales, and at low temperatures, physical matter acts in ways that are different to those of our everyday experience. The gas Helium, for example, when cooled to sufficiently low temperatures undergoes a change of state and becomes a superfluid, exhibiting highly unusual properties. Tiny droplets of superfluid Helium, in which different types of molecules have been dissolved, might in the future have a wide range of uses. Quantum Mechanics provides mathematical techniques that can be used to understand the behaviour of materials like this. The pictures illustrate a recent "Bosons on a Ring (BoaR)" model, in which a single central molecule is surrounded by a ring of Helium atoms. The work aims to answer the following question: since superfluidity is a property exhibited by "bulk" liquid Helium, rather than single atoms, how few atoms can be present in order for the phenomenon to occur?

### Solar sails

A Solar Sail is a novel type of spacecraft, which uses sunlight for its propulsion. Solar sails are capable of unusual orbits and missions which conventional spacecraft are not. Recent work has focused on the periodic orbits of solar sails in the Earth-Sun system, and in particular their stability.

### Statistical physics

The goal of Statistical Physics is to understanding how the interactions between the individual, sometimes microscopic, elements of physical systems leads to the emergence of large scale properties such as pattern, order and dramatic changes in state. Our recent work has focussed on understanding how collective behaviour of groups of agents playing games (competing for the best investments, searching for the best territory) can emerge from simple models of their intelligence. We also use ideas from statistical physics to understand how certain unusual rock formations appear, and why the distributions of landslide sizes and disease outbreaks may have a "double pareto" distribution due to shocks in the processes which drive them. This work has led to the discovery of some new families of Stochastic Jump Processes.

### Transitions to strange non-chaotic behaviour

Objects called strange non-chaotic attractors (SNAs) can arise in the study of quasiperiodically forced systems, and are fractal sets upon which nearby trajectories do not separate exponentially (on average). These seemingly paradoxical objects have been of great interest over the past three decades, after first being shown to exist in a paper by Grebogi et al. in 1984. Our recent contribution to this topic has been in the exploration of the “non-smooth pitchfork bifurcation”. In particular, we have examined scaling properties and the box-counting dimension of attractors arising from this route to SNA. One of the key ingredients in this analysis is the technique of renormalisation, which can be used under certain circumstances to show self-similarity of the attractors and to calculate scaling factors describing their growth near the point of transition.