2019 Hatherton Award: Mathematics of tipping points and more
Christian Offen, Massey University, has been awarded the Hatherton Award for his paper that develops a new framework for a class of non-linear differential equations that have values at which the number of solutions changes.
Royal Society Te Apārangi presents this award to the best scientific paper by a PhD student at any New Zealand university, studying chemical, physical, mathematical or information sciences.
Nonlinear systems of differential equations are the basis for modelling many problems in science and technology, including biological pattern formation, viscous fluid flow phenomena, chemical reactions and crystal growth processes.
Differential equations have been at the foundation of a precise understanding of the physical and technological world for centuries. These mathematical models usually depend on a number of parameters. Parameter values at which the number of solutions to the equations change are particularly important. These are referred to as bifurcation points. Tipping points, hysteresis (a phenomenon in which the value of a physical property lags behind changes in the effect causing it) and bistability—a system that has two stable equilibrium states—are all common phenomena associated with bifurcation points.
Christian's paper is concerned with the nature of bifurcations in a very wide class of problems, namely Hamiltonian boundary value problems. ‘Hamiltonian’ refers to the source of the equations in fundamental physical laws such as Newton's laws of motion; ‘boundary value problems’ are those in which the starting and finishing points are specified, such as a loaded beam pinned at both ends, or an orbital trajectory problem.
Using advanced geometry and analysis, Christian has created a new framework for studying these bifurcations. Christian's original approach is to phrase the question in terms of geometry, namely the intersection of special kinds of surfaces called Lagrangian submanifolds, and then to translate this to the language of ‘catastrophe theory’. He has found his framework useful for the general situation and it has allowed him to discover new features. For example, some common types of boundary conditions restrict which types of bifurcations can occur. In addition, he discovered a new phenomenon that he christened the ‘periodic pitchfork bifurcation’, in which one solution turns into three, passing through a periodic state.
On receiving this award, Christian said: “I am very pleased to receive the Hatherton Award for my paper on Hamiltonian boundary value problems. The important discovery which lead to the paper came early on during my PhD studies and it paid off to pursue the idea.
“The paper will be the central piece of my PhD thesis and provided a starting point for several other papers. I am grateful to my colleagues for all their support, especially my co-author and supervisor Robert McLachlan.”
Christian's initial research has now led on to five further papers, a testament to the novelty, degree of difficulty and wide application of his findings. Christian is currently at the Newton Institute at Cambridge University, UK.
For the best scientific paper by a PhD student at any New Zealand university in chemical sciences, physical sciences, mathematical and information sciences.
To Christian Offen for his paper outlining his definitive study of Hamiltonian boundary value problems with widespread application to bifurcations and tipping points
RI McLachlan and C Offen: Bifurcation of solutions to Hamiltonian boundary value problems, Nonlinearity 31 (2018) 2895–2927.