Gianne Derks

Gianne's research

On this page you can find information about the following topics.

For more information about similar research, see the following research group web pages

Research Interests

My research is in nonlinear dynamical systems in the areas of nonlinear wave equations, geometric mechanics, patterns and symmetry, and biological models. Recurrent topics in my research are slow-fast systems, solitons and fronts in symplectic and multi-symplectic systems, relative equilibria in Hamiltonian systems and effects of perturbations on those objects. Apart from tackling fundamental mathematical questions in those areas, I am also working on the mathematical modelling of cancer treatments, pharmacokinetic-pharmacodynamic (PKPD) behaviour of monoclonal antibodies, sleep-wake and circadian cycles, and AMR spread between humans and animals. Below are some of my research projects.

Inhomogeneous wave equations

The nonlinear wave equation utt=uxx+V(u) can have (travelling wave) front or solitary wave solutions. If the spatial medium is not homogeneous, then the nonlinearity will depend on x too. For example, one can look at a potential with V(u)=V1(u) for |x|>L and V(u)=V2(u) for |x|<L. This can lead to a plethora of stationary fronts or solitary waves if the length L of the inhomogeneity is considered as a parameter. Changes in the stability of those waves can be linked to extremal points of the length parameter as function of the "energy". Applications can be found in Josephson junctions but also in models for DNA/RNAP interaction.

Mathematical models in pharmacology

Attacting slow manifolds in pharmacokinetics-pharmacodynamics (PK/PD) models play an important role in getting a better understanding of interactions of drugs and targets. We investigate such manifolds in target-mediated drug disposition (TMDD) and Michealis-Menten (MM) models. We are also studying models for cancer growth and therapy as well as models describing anti-microbial resistance (AMR).

Slow-fast systems

Wave equations and models in pharmacology often have a slow-fast structure. We are using this structure and singular geometric perturbation theory to analyse existence of fronts and periodic waves as well as transitions in pharmacology models.

Sleep-wake dynamics and circle maps with discontinuities

We are working on understanding the dynamics underlying many of the sleep-wake models. We are analysing those models, using slow-fast systems and circle maps. This link has motivated us to investigate the more fundamental question of the creation of discontinuities in circle maps.

Multi-symplectic structures and solitons

Many Hamiltonian wave equations can be written as a multi-symplectic system. This means that the temporal and all spatial variables have a symplectic structure associated with it. This additional structure has several advantages. To analyse the stability of solitary waves or dimension breaking, one can use the socalled symplectic Evans matrix.

Effects of perturbations on Hamiltonian systems with symmetry

Hamiltonian systems occur often in modelling. They are dynamical systems of ordinary differential equations or partial differential equations with some extra structure. In physical models an example of a Hamiltonian system is a system which conserves the energy, like an undamped pendulum or a non-viscous fluid.
But in real life the nice structure of the Hamiltonian system is often slightly perturbed. For example, a pendulum does not keep swinging, but slowly gets damped. However, often one can observe features in the perturbed system, which look very much as in the Hamiltonian system.
In my research I want to find which features of the Hamiltonian system can help you in understanding and analysing the perturbed system. Topics that appear in this context are:

  • Approximations of solutions of Hamiltonian systems with dissipation;
  • Attracting sets in Hamiltonian systems with dissipation;
  • Persistence and/or bifurcation of travelling waves or other nice solutions in water wave problems;
  • Relation between blow-up in nonlinear Schrodinger equations and the behaviour of solutions in the generalised complex Ginzburg-Landau equations.

Dissipation versus Hamiltonian

Intuitively most people consider Hamiltonian systems to be the opposite of dissipative systems. The motivation is that a Hamiltonian system conserves quantities and a dissipative system makes quantities decay. However, if one looks at a system which can be seen as partly Hamiltonian and partly dissipative, then this picture is not so clear anymore. I am working to find criteria to split a system into a Hamiltonian part and a dissipative part.

Invariant Manifolds

There are nice theories and computer programs related to the continuation and bifurcation of one solution of a differential equation. However, doing a similar thing for invariant manifolds, is much less developed. So I want to look at the question when and how can we (numerically) continue manifolds and which kinds of bifurcations can we detect?

PhD thesis

My PhD thesis deals with coherent structures, called relative equilibria, in Hamiltonian systems with a non-Hamiltonian perturbation, like dissipation and/or forcing. If you like to read the summary of this thesis, click here.
Last modified: Sat Apr 27 16:43:38 BST 2019