Phylogenetic trees are commonly utilised in evolutionary biology. These trees represent the evolution of a set of species. In case of gene transfer however, a phylogenetic tree is insufficient. In such cases, tree-based phylogenetic networks are more suitable as they can depict reticulate evolution.
Nonetheless, tree-based phylogenetic networks have their limitations as they are unable to represent reticulate events occurring between different environments, families, or lineages of species. To address this, forest-based networks offer a solution.
A forest-based network is a collection of leaf-disjoint phylogenetic trees with arcs added between different trees in case of reticulate evolution.
In this thesis, an integer linear programme (ILP) is created to determine whether a binary phylogenetic network is forest-based. It is important to note that this thesis exclusively focuses on binary phylogenetic networks.
This ILP model is expected to be valuable in the development of new models and algorithms for both binary and non-binary phylogenetic networks, thereby enhancing the understanding of reticulate evolution. Additionally, the ILP may lead to time-related challenges when processing bigger phylogenetic networks.

The main aim of the research presented in this report is investigating analytical methods to model fluid-structure interaction in large-scale offshore floating photovoltaics. The model that was attempted to be solved analytically is based on a model presented by Pengpeng Xu (2022).
The dimensions in the equations were removed. Applying a perturbation method yielded hierarchic partial differential equations by introducing the wave amplitude divided by the depth of the ocean as a small perturbation parameter. The analytical solution of the first order problem was found by applying separation of variables and by using a Fourier transform. For certain classes of problems it is shown in this report that it is possible to analytically solve a model for fluid-structure interaction in offshore solar farms for various initial conditions.

Central configurations provide the only closed-form analytical solutions of the n-body problem. All possible central configurations of three bodies have been extensively studied along with the stability of the associated periodic orbits. Stable cases have been found for the Lagrangian triangle configuration, which we see occurring with the Trojan asteroids. However, the knowledge about four-body central configurations remains limited. An explicit parameterization of a family of kite shaped four-body central configurations has recently been published. The present research investigates the stability of periodic solutions provided by these central configurations. An analytical treatment of linear stability is carried out and the eigenvalues for circular periodic orbits are calculated. This is complemented with a numerical estimation of Floquet multipliers to determine the linear stability of eccentric periodic orbits. While most of the kite configurations are found to be unstable, regions of linearly stable cases are discovered for both circular and eccentric orbits. Further, numerical simulations of the non-linear system are performed as an independent approach to validate the linear stability results. Perfect agreement with the linear analysis is found, suggesting that stable kites may be observed in the universe.

The Ekman spiral is described by a coupled system of differential equations originally discussed by Walfrid Ekman (Ekman, 1905). This system is a simplified version of the Navier­Stokes equations. The differential equations, as discussed in Ekman’s paper, concern the currents of the ocean. However, it is also possible to interpret these equations so as to describe and predict the flow of wind. The research as presented is not only inspired by Walfrid Ekman’s original paper, but also by the master thesis from de Jong (2021). The main contribution of this thesis is to include the influence of a constant vertical wind speed on the classical Ekman spiral. After stuyding the classical Ekman spiral, the inclusion of a constant vertical wind speed is done step­wise. First, the vertical wind speed is discussed without having any vertical Coriolis forces. The classical Ekman spiral and the Ekman spiral with vertical wind, but no vertical Coriolis force, were solved exactly. Then, the vertical wind speed is included fully, giving rise to a non­linear coupled system of differential equations. For the non­linear system, an algorithm for solving it analytically using a general perturbation method is proposed. Next, the hodograph of the non­linear equations of motion including a constant vertical wind speed, is made using Euler’s Explicit numerical method and a shooting problem is solved.

Modelling natural phenomena via Dynamical System Theory has become commonplace amongst mathematicians, physicists, engineers, and the like. As such, research in this field is currently underway, with a notable focus on chaos. The Competitive Modes Conjecture is a relatively new approach in the field of chaotic Dynamical Systems, aiming to understand why a strange attractor is chaotic or not. Up till now, the Conjecture has only been used to study multipolynomial systems because of their simplicity. As such, the study of non-multipolynomial systems is sparse, filled with ambiguity, and lacks mathematical structure. This paper strives to rectify this dilemma, providing the mathematical background needed to rigorously apply a large set of non-multipolynomial systems to the Competitive Modes Conjecture. Examples of this new theory include application of Lorenz System, the Chua System, and the Wimol-Banlue System. As far as the authors are aware, any previous application of the latter two systems to the Conjecture has not been attempted. Therefore, this paper presents the first applications of a whole new set of dynamical systems to the Conjecture.

Graphene is expected to open up a world of new possibilities in the field of micro and nano sensors, due to its outstanding properties. However, graphene membranes suffer from high damping which limit their performance. The physical mechanism behind this damping is still unknown. Therefore, in this work, it is investigated how coupling between eigenmodes due to nonlinear stiffness causes damping. Besides, a method to construct this nonlinear stiffness for reduced order models is improved. Using this, numerical simulations are compared to experimental results for validation.

In this paper, it is analysed how an infinite cable behaves when a wave from the right hits a TMD system attached to it. The TMD system consists of a mass M1, a spring with stiffness K, a damper with coefficient C and a second mass M2. The cable has a tension T. Then it is investigated what happens when a wave goes through the cable and hits this TMD system. The goal of this paper is to calculate
how effective this TMD system damps. Using the multiple time scale perturbation method, the displacements of the cable and TMD system are calculated. This results in the reflected wave being damped by the TMD system, but the transmitted wave having a larger is placement than expected. The displacement of the TMD system also had unrealistic outcomes. To calculate the effectiveness of the TMD system, the energy which has been lost has to be calculated. Since the displacement of the TMD system had unrealistic outcomes, the calculation of
the energy is not realistic either.

In this research a study of the response of a simply supported microbeam subject to an electric actuation is presented. A perturbation method called the method of multiple scales is explained and used to solve our problem. A model concerning the mid-plane stretching and an electric force with a direct and alternating current component is formulated. The method of multiple scales is used to construct a solution that is valid for a long time after the initial conditions. The effect of the frequency of the alternating current was studied by performing a stability analysis. The results show that for frequencies close to the eigenfrequency of the homogeneous problem, there is no stable equilibrium and resonance occurs. Furthermore, a start was made to study the effect of the damping coefficient. The results show that a smaller damping will always lead to resonance on a very small time scale. The results also indicate larger oscillations and an small increase of the importance of the non-linear terms for smaller damping. All results are validated by comparing them with a numeric solution and show excellent agreement.

Cholera is an infectious disease caused by the bacterium Vibrio cholerae. Infections can occur directly via contact with infected people, but also indirectly via infected water. Because of these direct, but especially indirect infections, it is useful to investigate the spread of the disease using mathematical epidemic models. In this report a compartmental epidemic model for cholera is discussed: the SIRB model. This model consists of four compartments: susceptibles, infected, recovered and the bacterial concentration in water. The interaction between the groups is given by a system of differential equations. The equilibrium solutions and their stabilityare mathematically analysed. The main focus of this report, however, is on two different spatial models. The patches model consists of multiple patches connected by migration of people. Different measures against cholera are studied using this model. The second spatial model that has been numerically implemented is a diffusion SIRB model. The influence of the diffusion coefficients on the solution patterns will be researched for different initial conditions. The results presented in this report are building blocks for further research into compartmental models that simulate the spatial dynamics of cholera.

In this thesis the nonlinear spring system is considered. This system contains a semi-infinite string that is modelled by the wave equation with a pair of inititial conditions and a nonlinear boundary condition. The goal of this thesis is to find a good approximation of this system. Furthermore, the behaviour of the string is studied by plotting the reflected waves. Two methods are considered for estimating the solution. These are the Multiple Scales Perturbations method and the Fourth Order Runge Kutta method. The approximations of the two methods are compared to each other. From this, conclusions have been drawn on the accuracy of these approximations.

In this thesis the stability type of y=0 is being considered for the delay differential equation y''(t) + ay(t) + by(t-1) = 0 with a and b real numbers. It is already known that y=0 is stable when b=0 and a>0 and unstable when b=0 and a<0. The aim of this project is to determine the stability of y=0 for all values of a and b. First, the general stability theory for delay differential equations was highlighted before giving an in-depth stability analysis of the equation y''(t) + ay(t) + by(t-1) = 0. It turns out that a theorem of Pontryagin (1908 -1988) is really helpful for answering these stability questions. Due to this theorem all values for a and b are determined such that y=0 is asymptotically stable for y''(t) + ay(t) + by(t-1) = 0. However, this does not cover the stability type of y=0 for all values of a and b. So more analysis was done in order to give a full answer of the stability problem. The full answer was not achieved as there are still values for a and b where the stability is unknown. Finally, numerical solutions of y''(t) + ay(t) + by(t-1) = 0 are shown to confirm the results that are obtained.

Structure Optimization has been an important subject with many applications for centuries. In the last sixty years, numerical optimization has facilitated large advancements in this field. One of the areas in Structure Optimization is Topology Optimization, which is used for Additive Manufacturing purposes. In this thesis we explore Static and Dynamic Topology Optimization. In the optimization problems matrix equations of the type Ax = b, with A sparse and badly conditioned, are accelerated using deflation techniques in addition to preconditioning. We have applied several iterative methods, preconditioners, and deflation types to the topology optimization problems. The static problem concerns compliance optimization of a two-dimensional MBB-beam. The deflation type that reduced the number of iterations the most without introducing large overhead costs was rigid body modes deflation, divided over element squares. It was found that dividing the rigid body modes vectors over density based regions did not reduce the number of iterations. The dynamic problem concerns eigenvalue optimization of a three-dimensional moving wafer stage that is used for laser-printing computer chips. The optimization formulation contains a shifted eigenvalue problem that is solved using model order reduction. In the computations of bases for the reduction matrix equations appear, to which deflation techniques were applied. All deflation types reduced the number of iterations and needed time to solve the matrix equations. The best deflation type was using rigid body modes (RBM) divided over element cubes combined with eigenvectors from the previous iteration. The next best was the same combination without the division over cubes. Using eigenvectors or RBM separately were the least effective deflation types. There is an optimal amount of element cubes to use when dividing the RBM. The tests on a few grid sizes suggested a quadrupling of the amount when the grid size doubles, but more research is needed to really identify a relation between the grid size and optimal amount. When increasing the grid size to a level where parallel computing on a cluster was required, the deflation type using element cubes could not be used due to complications with parallel implementation. Using the deflation type RBM and eigenvectors, reductions by a factor of 1.60 and 1.75 in the total needed time for 150 optimization iterations were achieved for grid sizes 120x120x20 and 180x180x30, respectively. In the first case the objective function of the optimized design converged further in the same amount of iterations when using deflation. future research could include using the promising deflation type RBM in cubes + eigenvectors in parallel computing to obtain an even larger time reduction in the optimization of the wafer stage with large grid sizes.

Using an idealized width-averaged model, the influence of spring-neap cycles on the transport and trapping of suspended fine sediment in tide-dominated estuaries is investigated. To this end we introduce a multiple time scale expansion. This provides a mathematically sound argument for treating the fast ebb-flood cycle and the slow spring-neap variations as independent time scales. With this expansion, semi-analytical approximations of the water motion and suspended sediment concentrations can be found as functions of the spring-neap time scale. The advantage of this is that the transport of sediment is then investigated in a tidally-averaged sense using analytically obtained temporal dependencies and model simulations based on the conditions in 2005 of the Ems-Dollard estuary. We found that sediment import is strongly enhanced during spring tide and that a combination of vanishing import processes and sediment export due to river discharge resulted in a net export of sediment during neap tide. Furthermore the spring-neap varying deposition of fine sediment in a bottom pool on the river bed is investigated. We found that the time needed for the system to adjust to new equilibrium conditions can not be neglected within a spring-neap cycle as temporal lag effects are clearly visible in the dynamic behaviour of the bottom pool and suspended sediment concentrations. These temporal lag effects are sensitive to the choice of the parameter governing erosion. Assuming different values of the erosion parameter we have shown that the long-term characteristics of the bottom pool are sensitive to these temporal lag effects. Choosing a low value of the erosion parameter results in the presence of a bottom pool throughout the spring-neap cycle. With a high value of the erosion no bottom pool is formed at any time in the spring-neap cycle.

Ballistic capture is a transfer method which was first applied in 1990. It allows a spacecraft to approach a target celestial body and enter a (temporary) orbit around it without requiring manoeuvres in between. Ballistic capture is a promising concept, as it is expected to be safer, cheaper, and more flexible in terms of launch windows than a traditional Hohmann transfer. Currently, a computationally efficient method which simulateneously allows for an inisghtful description of the dynamics of the ballistic capture problem remains to be found. A potential solution lies within the field of Lagrangian Coherent Structures (LCS). LCS is defined as a separatrix of regions in a flow with distinct dynamics. It may be possible that LCS around a planet have some correspondence to results found using stable set manipulation, a classic technique for obtaining capture trajectories. In this research three new areas within the field relating LCS to ballistic capture are explored. Firstly, it has not yet been shown what LCS can be found in an area around a planet, without making use of a priori stable set information. Furthermore, it is unclear what the effect is of changing the integration time in the procedure of extracting LCS. Finally, there has not yet been an analysis to show how the LCS relate to stable sets with different number of revolutions n. In this work two algorithms for extracting LCS have been developed. One is based on the simple but efficient computation of the Finite Time Lyapunov Exponent (FTLE). Another is based on the more involved Variational Theory. Both algorithms are validated on a toy problem used frequently in LCS extraction studies, and are then applied to the Elliptic Restricted Three Body Problem (ERTBP). It is shown that LCS around a planet yield resemblance with stable set results. The FTLE-based algorithm is able to quickly and efficiently identify the shape of the stable set. The Weak Stability Boundary, however, can not be extracted distinctly. The Variational Theory-based algorithm yields more distinguishable results for the Weak Stability Boundary. It is shown that large and constant integration times are beneficial. It is shown that extracted LCS form an approximation of the average resulting WSB for all stable sets.

Mass-spring systems are commonly used in structural components. Understanding their characteristics and pitfalls is an important issue, combining physics and mathematics to prevent such systems from resonating and causing structures to weaken or collapse. This way, suitable solutions can be found such as the use of dampers with the right characteristics. In this paper we focus on such systems, in particular with a snap-through mechanism. Their dynamical behaviour and the in uence of a small disturbance on the mechanism, such as a wind force, are analyzed. A behavioral model is set up to which Melnikov's Method is applied, in order to analyze the behaviour numerically. The latter is done using the Trapezoidal method.

The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the softening material behaviour and the iterative nature of the Newton-Raphson type methods used in NLFEA. However, robust numerical simulations become increasingly important, for example due to recent developments in Groningen. 
To address these issues, sequentially linear analysis (SLA) was developed which exploits the fact that a linear analysis is inherently stable. By assuming a stepwise material degradation the nonlinear response of a structure can be approximated with a sequence of linear analyses. Although this approach has been proven to be effective for several case studies, the numerical performance is still a problem that has to be solved. After every linear analysis, a single element is damaged resulting in incremental damage. As a result, the system of equations only changes locally between these linear analyses. Traditional solution techniques do not exploit this property and calculate a matrix factorisation every linear analysis, resulting in high computational times per analysis step. Since SLA typically requires many linear analyses to obtain the desired structural response, this leads to unacceptable analysis times. The aim of this thesis is to improve the computational performance of SLA by developing numerical solution techniques which exploit the incremental approach of SLA. To this extend, the following methods have been developed.
A direct solution technique has been developed which is based on the Woodbury matrix identity. This identity allows for the numerically cheap computation of the inverse of a low-rank corrected matrix. In this approach, the expensive matrix factorisation does not have to be calculated every linear analysis step. Instead, the old factorisation can be reused along with some additional matrix- and vector multiplications and solving a significantly smaller linear system of equations. An optimal strategy is derived to determine at which point a new factorisation should be calculated.
An improved preconditioner for the conjugate gradient (CG) method has been developed. Instead of an incomplete factorisation, the complete factorisation is used as a preconditioner which reduces the number of required CG iterations significantly. The point at which too many CG iterations are required and a new factorisation is necessary, is determined using the same strategy as the first method. From numerical experiments it follows that both methods perform significantly better than the direct solution method, especially for large 3-dimensional problems. The best performance is achieved using the Woodbury matrix identity resulting in the solver no longer being the dominant factor in SLA. Furthermore, significantly larger problems are not solvable in time frames in which previously only smaller problems were solved.

In the past decades, several EOR (Enhanced Oil Recovery) techniques have been developed to increase the amount of oil recoverable from a reservoir. Among these techniques, polymer flooding consists in the injection of a solution of water and polymer into the reservoir, and it is performed in order to lower the mobility of injected water. A peculiarity of a water-polymer flow through a porous medium is the velocity enhancement, or hydrodynamic acceleration, that affects the polymer molecules. Experimentally, polymers are observed to travel faster than inert chemical species. Therefore, when simulating numerically a polymer flooding, a velocity enhancement effect has to be incorporated into the governing equations to accurately predict oil recovery. The simple model that introduces a constant enhancement factor, usually implemented in commercial simulators, leads to an ill-posed problem, causing stability issues in the simulations, which further leads to a monotonicity loss in the solution: the polymer is predicted to pile-up at the water-oil interface. While accumulation of polymer at the water front is not necessarily unphysical, it should not be the result of a mathematical ill-posed problem. Hence, an alternative model, employing a saturation dependent factor, has been proposed in the work of Bartelds et al. (G.A. Bartelds, J. Bruining, and J. Molenaar. The modeling of velocity enhancement in polymer flooding. Transport in Porous Media, 26(1):75–88, 1997), and has been extended by Hilden et al. (S.T. Hilden, H.M. Nilsen, and X. Raynaud. Study of the well-posedness of models for the
inaccessible pore volume in polymer flooding. Transport in Porous Media, 114(1):65–86, 2016) to overcome some physical restrictions. In this thesis, these models are re-examined through a thorough analytical study in the one-dimensional case. An analytical solution, resulting in an acceleration of the polymer front, is computed for the well-posed model proposed by Bartelds, while the extended enhancement factor derived by Hilden is shown to lead again to an ill-posed problem. A study of the monotonicity of the numerical schemes reveals that accumulation of polymer at the water front is strictly related to ill-posedness.
Obtaining accurate numerical solutions is not an easy task. Commercial simulators adopt fully implicit schemes to solve the multi-phase flow, and transport of chemical agents is solved through the development and coupling of separate modules. In this case, the use of high-resolution methods for the transport of polymer is not compatible with the first order solver for the underlying flow.
To investigate the interaction of the enhancement model proposed by Bartelds with other physical phenomena, adsorption of polymer onto the reservoir rock is added to the governing equations. The problem maintains the well-posedness property, but because of a raise in equations complexity, only numerical results are presented. The model is then extended to the two-dimensional case and, relying on numerical solutions, similar conclusions as in the one-dimensional case are derived.

In this project the transversal vibrations of an accelerating elevator cable system are studied, with the aim to find the resonance times, the resonance duration and the resonance amplitude. The elevator cable is modelled as an axially moving string, with length given by l(t) = l0 + 1/2 at2, with a the acceleration and t the time. The cable is sinusoidally excited at the top and fixed at the bottom. It is assumed that the axial acceleration is small compared to the transversal acceleration, that the cable mass is small compared to the car mass, and that the excitation amplitude is small compared to the length of the cable. Using these estimations, the solution for the transversal displacement u is approximated up to O(ε) with ε a small parameter. The elevator cable goes through a cascade of autoresonances: the eigenfrequencies of the cable are varying because the cable length is varying, and at several times an eigenfrequency matches the excitation frequency. These are the resonance times, and they have been found as t+ = (2/εa1l0)1/2arccos((Ωl0/χk)1/2), with t+ a measure of oscillation of t, Ω the angular excitation frequency, l0 the initial length, χk the eigenfrequency of mode k and εa1 = a. The duration of the resonances (the timescale) is shown to be O(ε-1/4) if χk≠Ωl0 and O(ε-1/6) if χk=Ωl0 (a bifurcation of the problem). The amplitude scale is thus O(ε3/4) or O(ε5/6), respectively, and solutions for the amplitude are calculated both outside and inside the resonance zone.   

In this report, the rain-wind induced vibrations of cables are studied. This is done by modeling the cable cross-section as a mass-spring system with two time-varying masses. Thereafter, the solution of this model is approximated using a multiple timescale perturbation method. Lastly, for some choices
of the time-varying masses the eigenfrequencies are analyzed, stability properties are derived, and approximations of the solutions are given.