Research Interests:

Statistical properties of dynamical systems:

Probability theory and statistics are of fundamental importance to the applied sciences. From weather prediction to analyzing genomic data, probability theory plays a crucial role. A probabilistic description of physical processes is also common in areas of physics such as statistical mechanics and thermodynamics. Although such systems may be described precisely by deterministic rules, like Newton's laws of motion, both the sheer number of interactions involved (think of the number of gas particles in a room) and the sensitivity of the prediction to slight errors in initial conditions (deterministic chaos) imply that a probabilistic description is often the most appropriate, and sometimes the only one possible.

Dynamical systems is the study of systems that evolve over time, for example the planets in the solar system change position and velocity over time in accord with Newton's laws of motion. Given the state of the solar system at time T, i.e. the velocity and position of the planets, the state can be predicted at time T+1 from Newton's laws of motion. This is just one setting in which there is a rule, in this case given by Newton's laws, which takes an initial state x to another state f(x) one unit of time later. We can consider the collection of all allowable initial states, call it the state space X and study the transformation f:X -> X. In this example X would be a set of possible positions and velocities for the planets. The dynamical point of view is to study the iterations x->f(x)->f(f(x)).... The n-th iterate of an initial condition x is usually written ,fⁿ (x). Is the motion periodic? chaotic? stable to perturbation? what is an efficient way to simulate the system numerically? Can we predict the state at time n with a certain degree of certainty, given that we can't know the initial state precisely? These are natural questions from this viewpoint.

If g : X -> R is a measurement on the system, for example the distance of a planet to the sun, then we can study how the time-series of observations { (x), (fx) , ..., (fⁿx) } behaves over time. In probability theory the sequence {X_n(x)= g (fⁿx)} would be called a stochastic process. If the stochastic process is independent (i.e. completely random) and identically distributed with finite second moments then classical theorems like the strong law of large numbers and the central limit theorem apply.

A main recent focus of my research has been the study of the statistics of time-series of observations {g(x), g(fx) , ...,g(fⁿx) } that arise from measurements on deterministic dynamical systems. The classical theories of probability usually do not apply, since these measurements are in general not completely random. There is a lot of memory in the system- if the outside temperature is 90F now then it is likely to be above 80F in half an hours time. In fact such time-series of measurements usually come from systems that are completely predictable- if we could know the exact initial condition precisely and calculate all variables involved. But in general this is impossible, due to extreme sensitivity to initial conditions and the number of particles and interactions involved. It is remarkable that a rigorous probabilistic understanding can be developed for deterministic physical systems and that it gives useful information and provides a deeper understanding of the physical world. For example extreme value theory is a a well-developed branch of classical statistics, and predicts the probability of extreme events like floods or hurricanes. We have recently extended this theory to models of chaotic physical systems and in particular the Lorenz equations, which were developed as a simplified model of the weather.

Equivariant dynamical systems and coupled cell systems:

Equivariant dynamics (or symmetric dynamical systems) studies the geometric and statisical properties of physical systems with symmetry. If a map or flow has, for example, a reflectional symmetry then this has strong implications for the mixing properties of the system and the possible symmetries of any attractors the map or flow may have. The goal in this area is to learn as much as possible about the behaviour of the system and the properties of the dynamics that it exhibits just by knowing the symmetries of the system. Some work in this area has described the behavior of spiral wave patterns in reactive media, such as cardiac tissue. In many physical and biological systems, many individual units are coupled together and the behavior of one affects the behavior of many. For example, neurons interacting in the cortex or transformers in an electrical supply grid. The study of such coupled systems is just developing, and although our understanding is limited, this is a very active area of research.

Dynamics of skew products and iterated function systems:

My other recent research interests have been in the dynamics associated with invariant subspaces, in particular mechanisms inducing on-off intermittent behaviour. Invariant subspaces or submanifolds often play an organisational role for the dynamics of the entire system. The main goal here is to analyse models which give rise to intermittency to produce a quantitative understanding of how the ergodic properties (the density of invariant measures, mixing properties, distribution of orbits, distribution of laminar phases) depend upon the parameters of the system. This work is of interest to engineers and physicists since it is basically the problem of understanding how and when systems stay synchronised. Understanding the synchronisation of coupled oscillators has been a major impetus to this line of research. Many of the mathematical models I have studied have a skew product structure. This work has applications to the study of Iterated Function Systems (IFS), which are used in computer graphics and data compression.

Non-stationary dynamical systems and stochastic modeling:

Recently I have been investigating dynamical systems which evolve over time or which are not in equilibrium and so time series arising from the system are not stationary processes. For example, non-stationary random walk models of phenomena where the jump increments depend on the size of previous jumps. Such models have been used to explain the currency exchange markets, which are not well modeled by standard random walks or Brownian motion. This has led to a study of the associated stochastic differential equations and Fokker Planck equations which also are used to model such systems.