Networks are often characterised in terms of their degree distribution and global clustering coefficient. It is assumed that these provide a sufficient parametrisation of networks. However, since the global clustering coefficient is only sensitive to the total number of triangles found in the network, it is evident that two networks could have the same number of triangles but significantly different higher-order structure, i.e., the topologies that result from the placement of closed subgraphs around nodes. The two main objectives of my work are: (1) developing network generating algorithms and network based epidemic models with controllable higher-order structure and (2) investigating the impact of higher-order structure on dynamics on networks. This thesis is based on three papers, corresponding to Chapters. 3, 4 and 5. Chapter 3 presents a novel higher-order structure based network generating algorithm and subgraph counting algorithm. Chapter. 4, generalises a previously proposed ODE model that accurately captures the time evolution of the susceptible-infected-recovered (SIR) dynamics on networks constructed using arbitrary subgraphs. Chapter. 5, improves, extends and generalises the network generating algorithms proposed in the previous two papers. All three chapters demonstrate that for a fixed degree distribution and global clustering, diverse higher-order structure is still possible and that this structure will impact significantly on dynamics unfolding on networks. Hence, we suggest that higher-order structure should receive more attention when analysing network-based systems and dynamics.

This work is focused on the approximation of sets of attractive solutions of planar dynamical systems. Existing work has shown that for many dynamical systems a Riemannian contraction metric can be used to determine sets of solutions with certain attraction properties. For autonomous dynamical systems in R² it has been shown that the Riemannian contraction metric can be reduced to a scalar weight function W. In this work we show that a similar result holds true for finite-time dynamical systems with one spatial dimension. We show how meshless collocation can be used to construct an approximation of W. The approximated weight function can then be used to determine subsets of the area of exponential attraction. This is the first time a method has been introduced to approximate finite-time areas of exponential attraction. We also give a convergence proof for the method. For autonomous dynamical systems in R² there already exists a method that uses W to determine a subset of the basin of attraction of an exponentially stable periodic orbit, Ω. However that method relies on properties of Ω being known. We show that the existing equation for W can be manipulated so that no knowledge of the periodic orbit is required to approximate W. We present a method that utilises meshless collocation to approximate W and show that the method is convergent. The approximant of W is then used to determine subsets of the basin of attraction of Ω.

Lyapunov functions are functions with negative orbital derivative, whose existence guarantee the stability of an equilibrium point of an ODE. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. In this thesis, we improve an established numerical method to construct Lyapunov functions using the radial basis functions (RBF) collocation method. The RBF collocation method approximates the solution of linear PDE's using scattered collocation points, and one of its applications is the construction of Lyapunov functions. More precisely, we approximate Lyapunov functions, that satisfy equations for their orbital derivative, using the RBF collocation method. Then, it turns out that the RBF approximant itself is a Lyapunov function. Our main contributions to improve this method are firstly to combine this construction method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of collocation points needed to construct Lyapunov functions. Moreover, we design two modified refinement algorithms to deal with the issue of the early termination of the original refinement algorithm without constructing a Lyapunov function. These algorithms uses cluster centres to place points where the Voronoi vertices failed to do so. Secondly, we derive two verification estimates, in terms of the first and second derivatives of the orbital derivative, to verify if the constructed function, with either a regular grid of collocation points or with one of the refinement algorithms, is a Lyapunov function, i.e., has negative orbital derivative over a given compact set. Finally, the methods are applied to several numerical examples up to 3 dimensions.