The field of Dynamical Systems is concerned with processes that evolve in time, particularly those processes that have nonlinear components. Dynamical systems reaches deeply into many areas—applied mathematics, pure mathematics, statistics and computational science—and is necessary to understand a wide range of complex natural phenomena. The research interests of the group cover arithmetic dynamics, differential and difference equations, ergodic theory, fractional calculus, hybrid discrete/continuous time equations, integrable systems, operator theory, and mathematical physics.

### Group Members

- Christopher Angstmann
- Gary Froyland
- Chris Goodrich
- Jonathan Kress
- John Roberts
- Wolfgang Schief
- John Steele
- Chris Tisdell

### Research Interests

**Christopher Angstmann** is interested in stochastic systems, random walks, and fractional calculus. Surprisingly, all three are connected. He explores these interests in applications across a range of fields including, physics, chemistry, biology, and finance.

**Gary Froyland** is interested in ergodic-theoretic, operator-theoretic, and differential-geometric approaches to dynamical systems. His recent work concerns random or time-dependent dynamical systems, and the use of transfer operator and Laplace operator constructions to analyse the statistics and geometry of these systems. He applies his research to solve dynamical problems in fluid flow and geophysical flows using models and data.

**Chris Goodrich** is interested in nonlocal differential and difference operators and their application in ODEs, PDEs, and difference equations. As part of this research he studies fractional calculus as well as boundary value problems which contain nonlocal terms such as Kirchhoff-type equations. His interests also include regularity theory for integral functionals and elliptic PDEs, particularly those containing VMO-type coefficients

**Jonathan Kress** is interested in superintegrable Hamiltonian systems and, in particular, using differential geometry, algebraic geometry and representation theory to methods to study and classify these systems.

**John Roberts** studies discrete integrable dynamical systems: their construction, their geometry and their properties. He is also interested in arithmetic dynamics -- the intersection of number theory with dynamical systems -- and, in particular, the dynamics of (bi)rational maps over discrete or finite number sets. Finally, he is also interested in transformation semigroups where the traditional dynamical idea of iterating one map is generalised.

**Wolfgang Schief**'s areas of interest are integrable systems, differential geometry and mathematical physics. In particular, he is concerned with the detection and utilisation of integrable structure in differential geometry and mathematical physics, and the development of a discrete analogue of differential geometry and its relation to discrete integrable systems.

**John Steele** is interested in General Relativity, particularly in exact solutions of the Einstein Field Equations, their symmetries and interpretation. He is also interested in geometric aspects of mathematical physics and the history of mathematical physics.

**Chris Tisdell** is currently kept awake at night by challenges involving the Navier-Stokes equation. It has not yet been proven whether solutions always exist in three dimensions and, if they do exist, whether they are smooth - i.e. they are infinitely differentiable at all points in the domain. The Clay Mathematics Institute has identified this as one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample. More broadly, Chris's mathematical interests involve nonlinear differential equations and the nature of their solutions. The methods involve new advances and applications of nonlinear analysis, difference equations and fixed-point theory.

### Research Fellows

Jason Atnip

### Relevant Courses

MATH3041 Mathematical Modelling for Real World Systems

MATH3201 Dynamical Systems and Chaos

MATH5175 Integrable Systems, Classical Theory and Modern Applications

### Recent Former Members

Bruce Henry