# Professor Gianne Derks

## Academic and research departments

Department of Mathematics, Centre for Mathematical and Computational Biology.### Biography

### Biography

I graduated as an "Ingenieur in Mathematics" at the Technical University of Eindhoven in 1988 and I got my PhD from the University of Twente in 1992. In 1993/94 I was awarded a NATO Research Fellowship to do research at the University of California Santa Cruz and the MSRI in Berkeley. In the following year 1994/95, I was a Research Fellow at the Simon Fraser University in Vancouver. In 1995, I started at the University of Surrey and I am currently a Professor in the Department of Mathematics.

### Research interests

- Hamiltonian systems (ODEs and PDEs) with perturbations like dissipation and/or forcing;
- Dissipation induced instability;
- Inhomogeneous wave equations;
- Stability and instability of travelling waves and front solutions, for example for (semi)-kinks in Josephson junctions or ionization waves in gas discharges;
- Multi-symplectic systems;
- Mathematical models in pharmacology, toxicology and metabolism (including pharmacokinetics-pharmacodynamics and physiology based dynamical models);
- Cycles in sleep-wake models, endocrinology;
- Wave dynamics in DNA-RNAP interactions;
- I'm a member of the LMS sponsored Network on Applied Geometric Mechanics;
- I'm a member of the LMS sponsored Network on Mathematics in Life Sciences;
- I'm on the organising committee of the Quantitative Systems Pharmacology-UK (QSP-UK) network;
- I'm on the editorial board of the Journal of Geometric Mechanics.

Further details can be found on my personal web page.

A full list of my publications and pre-prints can be found on my personal web page.

### Teaching

In 2019/20 I am lecturing the following module:

- MAT2007 Ordinary Differential Equations for level 2 Mathematical Studies students in Semester 1.
- MAT3046 Game Theory for level 3 Mathematical Studies students in Semester 1 (jointly with Prof Anne Skeldon).

Some suggestions for final year projects or literature reviews can be found here.

### My publications

### Publications

= u

+ V

(u, x) - ± u

(± e 0). The spatial real axis is divided in intervals I

, i = 0, ..., N + 1 and on each individual interval the potential is homogeneous, i.e., V (u, x) = V

(u) for x ? I

. By varying the lengths of the middle intervals, typically one can obtain large families of stationary front or solitary wave solutions. In these families, the lengths are functions of the energies associated with the potentials V

. In this paper we show that the existence of an eigenvalue zero of the linearisation operator about such a front or stationary wave is related to zeroes of the determinant of a Jacobian associated to the length functions. Furthermore, the methods by which the result is obtained is fully constructive and can subsequently be used to deduce the stability and instability of stationary fronts or solitary waves, as will be illustrated in examples. © 2012 Elsevier Inc. All rights reserved.

structure -- with two-form denoted by Omega - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts.

We introduce the concept of the "symplectic Evans matrix",

a matrix consisting of restricted "Omega-symplectic" forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter lambda to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large $|\lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case.

The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.

By combining all these results, a new rigorous sufficient

condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.

modes due to the discontinuities present in the system, it is shown that Josephson junctions with phase shift can

be an ideal setting for studying localized mode interactions. A phase-shift configuration acting as a double-well

potential is considered and shown to admit mode tunnelings between the wells. When the phase-shift configuration

is periodic, it is shown that localized excitations forming bright and dark solitons can be created.

Multimode approximations are derived confirming the numerical results.

Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is

a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity

of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that

the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold

with infinite codimension in an appropriate space of potentials.

standing of sleep/wake regulation. This ostensibly simple model is an interesting example

of a nonsmooth dynamical system whose rich dynamical structure has been relatively un-

explored. The two process model can be framed as a one-dimensional map of the circle

which, for some parameter regimes, has gaps. We show how border collision bifurcations

that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle-

node bifurcation set that is a feature of continuous circle maps. The novel picture that

results shows how the periodic solutions that are created by saddle-node bifurcations in

continuous maps transition to periodic solutions created by period-adding bifurcations as

seen in maps with gaps.

The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. These problems may exhibit oscillatory instabilities where eigenvalues detach from the edges of the continuous spectrum - so-called edge bifurcations. A numerical framework, based on a fast robust shooting algorithm using exterior algebra, is described. The complete algorithm is robust in the sense that it does not produce spurious unstable eigenvalues. The algorithm allows us to locate exactly where the unstable discrete eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows for stable shooting along multidimensional stable and unstable manifolds. The method is illustrated by computing the stability and instability of gap solitary waves of a coupled mode model.

We consider a spatially nonautonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-pi Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The nonautonomous character is due to the presence of a discontinuity point, namely, a jump of pi in the sine- Gordon phase. The continuum model admits static solitary waves which are called pi-kinks and are attached to the discontinuity point. For small forcing, there are three types of pi-kinks. We show that one of the kinks is stable and the others are unstable. There is a critical value of the forcing beyond which all static pi-kinks fail to exist. Up to this value, the (in)stability of the pi-kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of 2pi-kinks and -antikinks. Besides a pi-kink, the unforced system also admits a static 3pi-kink. This state is unstable in the continuum models. By combining analytical and numerical methods in the discrete model, it is shown that the stable pi-kink remains stable and that the unstable pi-kinks cannot be stabilized by decreasing the coupling. The 3pi-kink does become stable in the discrete model when the coupling is sufficiently weak.

The spectral problem associated with the linearization about solitary waves of spinor systems

or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans

function, a complex analytic function whose zeros correspond to eigenvalues. These problems

may exhibit oscillatory instabilities where eigenvalues detach from the edges of the continuous

spectrum, so called edge bifurcations. A numerical framework, based on a fast robust shooting

algorithm using exterior algebra is described. The complete algorithm is robust in the sense

that it does not produce spurious unstable eigenvalues. The algorithm allows to locate exactly

where the unstable discrete eigenvalues detach from the continuous spectrum. Moreover, the

algorithm allows for stable shooting along multi-dimensional stable and unstable manifolds.

The method is illustrated by computing the stability and instability of gap solitary waves of

a coupled mode model.

Techniques from singular perturbation theory such as asymptotic analysis, geometric singular perturbation theory and geometric desingularisation work well for the TMDD model where the underlying critical manifold consists of two two-dimensional submanifolds. However, the known techniques can not be applied to the dimerisation problem, as the underlying critical manifold is degenerate and consists of a two-dimensional and a three-dimensional submanifold.

The intersection of these manifolds is important for the Dimerisation problem as there is a type of rebound in the dimerisation problem occurring at the specified bifurcation point.

Motivated by the dimerisation problem, we consider a general two parameter slow-fast system in which the critical manifold consists of a one-dimensional and a two-dimensional submanifold. These submanifolds intersect transversally at the origin.

Using geometric desingularisation, we show that for a particular subset of parameters the continuation of the slow manifold connects the attracting components of the critical set. We also show that the direction of this continuation on the two-dimensional manifold can be expressed in terms of the model parameters.

This method is then applied to the dimerisation problem in order to understand and approximate the rebound.

interaction between two oscillators. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively unexplored. A key aim of this work is to further understand how transitions between monophasic (one sleep a day) and polyphasic (many sleeps a day) sleep occur in the two process model.

The two process model can be framed as a one-dimensional map of the circle which, for some parameter regimes, has gaps. As is a feature of continuous circle maps the bifurcation set consists of saddle-node Arnold tongues. We show that border collision bifurcations that arise naturally in maps with gaps extend and supplement these tongues. We see how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps. With this deeper understanding of the dynamics and bifurcation structure of the two process model we use modifiedý versions of the model to explain two experimental data sets.

An ultradian rhythm is a recurrent period or cycle which repeats multiple times across the day. We consider the sleep wake patterns of a the common vole, Microtus Arvalis, which has ultradian rest activity and feeding patterns. By deriving parameters for the two process model from EEG data and sleep/ wake onset times we are able to simulate with high accuracy the key features of spontaneous sleep-wake patterns in the voles. However, to explain phenomena seen in sleep deprivation experiments we include a high amplitude ultradian oscillation alongside the circadian, the results allow us to give some physiological insight into the internal mechanisms which drive sleep/wake onset times in the common vole.

Across the human lifespan there are many changes in the physiological properties of sleep, sleep timing and sleep duration. In adolescence sleep timing is delayed and there is a reduction in slow wave sleep which continues into old age as sleep timing gradually becomes earlier. Using a modified two process model which incorporates a van der Pol oscillator driven by external light signals into the circadian process we show that changes in sleep timing and duration across the lifespan can be explained by varying parameters. Model simulation show that these changes can be understood by a simultaneous reduction in the amplitude of the circadian oscillator and the upper asymptote of the homeostatic sleep pressure.

a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous

sine-Gordon equation has stable stationary front solutions that persist in the coupled system.

Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts

loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is

strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating

fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function.

With this approximation, we prove analytically the existence of a pitchfork bifurcation. To

complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity

persist for the smooth ?hat-like? spatial inhomogeneity by introducing a fast-slow structure and

using geometric singular perturbation theory.

The uniformly damped Korteweg--de Vries (KdV) equation with periodic boundary conditions can be viewed as a Hamiltonian system with dissipation added. The KdV equation is the Hamiltonian part and it has a two-dimensional family of relative equilibria. These relative equilibria are space-periodic soliton-like waves, known as cnoidal waves. Solutions of the dissipative system, starting near a cnoidal wave, are approximated with a long curve on the family of cnoidal waves. This approximation curve consists of a quasi-static succession of cnoidal waves. The approximation process is sharp in the sense that as a solution tends to zero as *t* to infinity, the difference between the solution and the approximation tends to zero in a norm that sharply picks out their difference in shape. More explicitly, the difference in shape between a solution and a quasi-static cnoidal-wave approximation is of the order of the damping rate times the norm of the cnoidal-wave at each instant.