Dynamical systems and complexity
Understanding the temporal behaviour of dynamical systems defined by differential equations or maps is key to many questions in applications as well as in Pure Mathematics. Our work studies a number of fundamental problems and phenomena. The weekly cross-disciplinary seminar series organised by the Centre for Networks and Collective Behaviour generates lively interdisciplinary discussion and supports extensive contact between Mathematical Sciences and many other research groups across the University.
Our research activities include the following themes:
- Ergodic theory
- Gradient flows
- Many particle systems and lattice dynamical systems
- Mathematical chemistry
- Non-smooth dynamics
- Pattern formation and symmetric bifurcation theory
Related areas of our research include:
- Analysis and Differential Equations
- Continuum Mechanics of Solids and Fluids
- Industrial Applied Mathematics
- Mathematical Biology
- Mathematical Control Theory
- Numerical Analysis
Further details are given below.
Research staff: Anthony Dooley
Research in Ergodic Theory on non-singular measurable dynamical systems has included development of the theory of G-measures: it can now be shown that they all have the structure of Bratteli-Vershik systems with Markov odometer actions. This opens up new questions on the structure and classification of dynamical systems. The theory of critical dimension for non-singular systems controls how quickly the Radon-Nikodým derivative sums grow along orbits, and is linked to a notion of non-singular entropy.
Systems of completely positive entropy with amenable group exist in abundance, and continue to be objects of investigation. Current interests also include probability on groups and the theory of random dynamical systems.
Nonequilibrium problems and the derivation of thermodynamic quantities, such as entropy, from microscopic models are of considerable current interest. A starting point is here the analysis of the over-damped limit of an evolution, that is, a context where the damping is so strong that inertial effects can be neglected. Then the behaviour of many materials can be described by the desire to reduce their energy. We call an evolution a gradient flow when, loosely speaking, a dynamical system moves at any time in the direction of the steepest descent of the energy. Here a fascinating problem is to derive rigorously the effective energy and evolution equation for a given microscopic model. For example, it is known that the diffusion equation can be interpreted as gradient flow of the entropy in the Wasserstein metric. How can we link this description directly to an underlying model of Brownian motion? This link shows that entropy is indeed macroscopically the driving force, and explains why the Wasserstein metric occurs. The argument combines probabilistic and analytic techniques, namely Large Deviation Principles and Gamma-convergence.
Dynamical systems with many degrees of freedom occur when physical systems are resolved up to the atomic level. Particles form then lattices or gases. Recent research studies the connection and differences of such discrete systems and the corresponding continuum descriptions via partial differential equations.
Many physical problems are well described on a microscopic scale by atoms which are linked by nonlinear springs. Such models appear in solid state physics (e.g., in models for the elastic and plastic behaviour of crystals). The governing equations are so-called lattice dynamical systems; these systems arise as well in statistical mechanics and biology, for example as idealisations of DNA molecules in the biological sciences.
Fermi, Pasta and Ulam discovered in 1953 the complexity of these systems in a now-famous computer simulation of a chain of 64 atoms. The discovery that the system does not behave ergodically strongly influenced the development of KAM theory, chaos and solitons. Many challenges remain; one important problem is the existence theory for travelling waves in lattice dynamical systems. A variety of tools from the calculus of variations and PDEs can be applied, such as mountain pass arguments or compensated compactness. Yet, many tools from PDEs cannot be easily adapted, and the existence theory for lattice dynamical systems is far from satisfactory. One challenge is to prove the existence of waves for a number of lattice dynamical systems; another challenge is to prove qualitative statements, such as stability.
The dynamics of atoms and molecules pose challenging mathematical problems. Many problems can be described as a potential energy landscape with many potential wells separated by barriers. One then often wants to find a trajectory joining a given initial configuration with a given final one. For example, the two configurations could be different conformational states of a molecule. The dynamics is complicated: Typically, the trajectories will jostle around in the well belonging to the initial configuration, before a rare spontaneous fluctuation occurs and the trajectory crosses the barrier and reaches the next valley of the energy landscape. How can we compute these rare events?
Research staff: Chris Budd
Classical work on dynamical systems assumes that the underlying differential equations or maps are smooth. However, many problems in applications do not satisfy this assumption. These include problems with impact, friction, sliding and switching. In Bath there is significant interest in studying the basic principles of such non-smooth dynamical systems and in investigating and classifying the possible forms of the dynamics. This involves studying novel types of bifurcation (such as grazing, border-collision and sliding) and the subsequent behaviours, such as the period-adding, period incrementing and anharmonic routes to chaos. We also work closely with members of the departments of mechanical and electrical/electronic engineering to study applications in bearing dynamics and in switching networks.
Pattern formation is the study of the spontaneous appearance of spatial structure, often with a characteristic wavelength, in externally driven, dissipative nonlinear PDEs and spatially discrete coupled cell systems.
Spatial structure appears through symmetry-breaking instabilities (bifurcations) and symmetry plays a central role in organising the new solution branches that appear. In some cases the system supports localised patches of pattern, and there is a rich bifurcation structure that explains aspects of their behaviour, at least in one spatial dimension. Research on these, and many other aspects of dissipative PDEs, typically involves a mixture of numerical and asymptotic analysis together with an appreciation of how bifurcation-theoretic ideas are augmented by methods involving group theory.
Spatial structures can also be forced by an underlying heterogeneous medium. A way to study particular elliptic equations on infinite strips is to treatthe unbounded direction as if it were "time", and to interpret the equations dynamically. The behaviour of travelling waves in inhomogeneous media can be understood much better by combining spatial dynamics with averaging and homogenisation. The effect known as "pinning" is of special interest.